extra problems for jacobs using gsp

Extra Exercises for Jacobs using Geometer's Sketchpad

All of these exercise are adapted from exercises in Geometry: Seeing Doing, Understanding

(Third Edition) by Harold R. Jacobs. They are designed for an instructor who wants to make

fairly heavy use of Geometer's Sketchpad (Version 5), while using Jacobs' excellent text.

Exercises are arranged by section and I have noted in square brackets what exercises they are

intended to replace. Occasionally I direct the student to Jacobs for a diagram or extra details, but

most exercises can be done without reference to Jacobs.

Section 1.4 [Set III 1 – 7]

1. Select Graph > Define Coordinate System. Then select Graph > Snap Points. Select the Point

Tool (from the left menu) and click on (8,6). Select the Translation Arrow Tool and with the

point (8,6) still selected, select the x-axis and then select Construct > Perpendicular Line.

Select the x-axis again and select Construct > Intersection, which should create a point at

(8,0). Deselect all by clicking in an empty space. Now select (8,6) and the y-axis and select

Construct > Perpendicular Line. Select the y-axis again and select Construct > Intersection,

which should create a point at (0,6). Deselect all, and select the two perpendicular lines that

you just created, then select Display > Hide Perpendicular Lines. Then select, in order, the

points (0,0), (8,0), (8,6) and (0,6) and select Display > Label Points with First Label = A and

click OK. Then select Construct > Segments. You should now have a rectangle ABCD

drawn. Deselect all.

2. Bisect A∠ by selecting D, A, and B in that order and then Construct > Bisect Angle.

Deselect all. In a similar fashion bisect angles B, C, and D. The four rays form a polygon

inside the rectangle. What type of polygon does this seem to be? Put your answer to the

question right in the sketch by clicking on the text tool, dragging to create a text box, and

typing in the text box. Save the sketch as <your_name> 1-4-2.

3. Drag the point C to (9,6). What is different about the polygon formed in this exercise and the

polygon in Exercise 2? What is the same? Add your answer to the text box.

4. Drag the point C to (8,4). Describe any changes in the text box.

5. Drag the point C to (8,2). Compare your results with the previous three cases.

6. How is the size of the polygon formed related to the shape of the rectangle? Type your

answer in the text box.

7. Do you think there is any rectangle for which the polygon would shrink to a point? Explain.

Type your answer in the text box. Save the sketch.

Section 1.5 [14 – 15]

1. Open a new sketch and create an equilateral triangle with side lengths 6 as follows: Select

Graph > Define Coordinate System. Then select Graph > Plot Points and plot the point with

coordinates (6,0) by using either Graph > Plot Points or Graph > Plot Value on Axis (x).

Select (0,0) and Transform > Mark Center. Select the point (6,0) and Transform > Rotate

60°. Deselect all, then select points (0,0), (6,0) and the new point in that order, and select

Display > Label Points with First Label = A. Then select Construct > Segments. You should

have an equilateral triangle ABC. Trisect (divide into equal parts) segment AB by plotting

points (2,0) and (4,0). Label these points D and E, in that order. Construct segments CD and

CE.

2. Deselect all. Select points A, C, and D in that order, and then click Measure > Angle.

Deselect all. Select points D, C, and E in that order, and then click Measure > Angle.

Deselect all. Select points E, C, and B in that order, and then click Measure > Angle. Do all

three angles appear equal? Do two of the three appear equal? Save the sketch as

<your_name> 1-5-2.


Read the three paragraphs under the heading Set III on page 97 for the interesting background of

this problem. The figure there is reproduced below. The numbers are the angle measurements in

degrees, and you are asked to find (by measurement) BFE∠ .

1. Make a sketch of the figure as follows. Do not use a coordinate system. First draw AB by

using the Segment Straightedge Tool. Make it any convenient length, perhaps between 1 inch

and 2 inches. Select the tool, click on the canvas, and hold down the shift key as you drag to

the right. (If you don't down the shift key, the line will not be exactly horizontal.) Deselect all

and then Select A and B in that order and select Construct > Ray to create the ray AB.

Deselect all and then Select B and A in that order and select Construct > Ray to create the ray

BA. Rotate ray AB 60° around A and then rotate the rotated ray 20° further.

? 30

40

10

70

20

60

C

G

F

E

A B

Rotate ray BA −70° and then −10° further around point B. (A positive rotation is

counterclockwise, negative is clockwise.) Select the two outside rays and Construct >

Intersection. Now choose the points labeled A, B, and C in the book diagram in that order,

and label them starting with A. Select Construct > Segments to draw triangle ABC. Deselect

all. Select segment BC and the inner ray starting at A, and Construct > Intersection. Deselect

All. Select AC and the inner ray starting at B and Construct > Intersection. Hide all 4 rays.

(To avoid hiding the segments by mistake, click on the part of each ray that is outside ABC�

.) Label the point on BC as E, and the point on AC as F. Construct segments AE and BF, and

then construct their intersection. Label it G. Construct segment EF.

2. Without measuring, you should be able to figure out the measures of the 4 angles with vertex

G, as well as AFB∠ and ACB∠ . Do so, and explain your reasoning in a text box. Use

Measure > Angle to confirm your answers.

3. Measure BFE∠ . This is very likely the answer to the problem. Save the sketch as

<your_name> 3-3-3.

Section 4.6 [15 – 25]

In Sketchpad you can bisect an angle using the Construct > Bisector command. However, in

these exercises you are asked to do it without this command, in the same way that you would do

it with compass and straightedge on paper.

1. Open a new sketch. Draw an angle ACB of about 50°, where CA and CB are segments .

Using the compass tool draw a circle with center C and radius chosen so that it intersects CA

(in D) and CB (in E).

2. Use the compass tool to draw a second circle centered at D that is large enough to reach more

than half way to CB. Construct a radius of this circle by selecting points D and the unlabeled

point on the circle and selecting Construct > Segment.

3. Construct a third circle of the same size as the second by selecting point E (with the radius of

the second circle still selected) and then selecting Construct > Circle by Center + Radius.

Then select the second and third circles and select Construct > Intersections. At least one of

the intersections will be inside ACB∠ . Label it F. (If both intersections are inside the angle,

label either one of them F.)

4. Construct ray AF. This is the angle bisector of ACB∠ . To verify this, measure angles ACB,

ACF, and FCB and note that 1

ACF FCB ACB2

∠ = ∠ = ∠ . (To measure an angle, you must

select three points, with the first and third selected points on the two sides of the angle, and

the second point selected at the vertex, then select Measure > Angle.) Save your sketch as

<your_name> 4-6-4.

In the next group of exercises, you are asked to show why the above construction works.

5. Select Save As and save the sketch in Exercise 4 as <your_name> 4-6-7. Select points D and

F and Construct > Segment. With the segment still selected, choose Display > Color and

choose a contrasting color, say red. Linear objects you create from here on will be in this new

color until you change the color.

6. Construct segment DE. The two colored segments are auxiliary lines, used in the proof but

not in the construction.

7. Construct a text box, and write the proof by copying each of the following statements into the

box, and giving a reason for each. To type in an equation or a math symbol that is not on

your keyboard, use the text bar that appears when you open the text box, selecting the button

2

3

πand, if necessary, click the down arrow on the right side of the text bar to see more

symbols. Save the sketch.

a. CD = CE

b. DF = FE

c. CF = CF

d. CDF CEF≅� �

e. DCF ECF∠ ≅ ∠

In the next group of exercises, you will construct the perpendicular bisector of a line segment

without using Construct > Midpoint and Construct > Perpendicular Line, but doing it the way

you have done it with compass and straightedge.

8. Start a new sketch. Draw a line segment AB of any convenient length.

9. Construct a circle with center A passing through B.

10. Construct a circle with center B passing through A.

11. Find the intersections of the two circles, and label them C and D.

12. Construct the line through C and D. This is the perpendicular bisector of AB.

13. Find the intersection of segment AB and line CD. This is the midpoint of AB.

In the next group of exercises, you are asked to show why the above construction works.

14. Construct auxiliary segments AB, BD, AC, and BC. Give each of these segments a red or

another distinctive color.

15. Open a text box, and copy the following statements, together with the reason(s) you know

each is true.

a. ACD BCD≅� �

b. ACD BCD∠ ≅ ∠

c. ACD BCD≅� �

d. AE = BE

e. AEC CEB∠ ≅ ∠

f. AEC 90∠ = °

g. CD is the perpendicular bisector of AB

Save the sketch as <your_name>4-6-16.

Section 4.7 [13 – 21]

1. Do problem 13 on page 173 using Sketchpad. Include a text box with a description of what

you did. Save your sketch as <your_name> 4-7-1.

2. Start a new sketch. Construct a line segment XY near the middle of the window. Select the

segment and select. Construct > Midpoint. Label the midpoint M. With the midpoint

selected, select segment XY and select Construct > Perpendicular Line to create the

perpendicular bisector of XY. Deselect all and then select M and X and select Measure >

Distance. Deselect all and then select M and Y and select Measure > Distance. You should

find that X and Y are the same distance from M. (After all, that is what we mean by the

midpoint.) Open a text box and type d(X, M) = d(Y, M) in the box. This is a shorthand way

of saying that the distance from X to M is the same as the distance from Y to M. Other ways

of expressing the same fact are to say X and Y are equidistant from M, XM and YM are the

same length, or XM = YM.

3. Deselect all and then select the perpendicular bisector you created in the last exercise. Select

Construct > Point on Perpendicular Line. Repeat this twice so you have created 3 points on

the perpendicular line. If any of the points are very close together, drag the points to separate

them. Sketchpad will constrain them to stay on the line. Now select the 3 points in top to

bottom order, and select Display > Label Points, with the first point labeled A. Next deselect

all and select A and Y and select Measure > Distance. Deselect all and select A and X and

select Measure > Distance. Your measurements should show that d(X, A) = d(Y, A). Type

d(X, A) = d(Y, A) in the text box. Repeat this with A replaced by B and C.

4. How many points do you think can be found equidistant from X and Y? Type your answer in

the text box.

5. Where do you think that all points that are equidistant from X and Y are located? Type your

answer in the text box, and save the sketch as <your_name> 4-7-5.

Chapter 4 Summary & Review [35 – 36]

1. Open a new sketch, and make a sketch consisting of four lines l, m, n, and o intersecting in 6

points A, B, C, D, E, F like this:

Suggestion: The exact measurements are unimportant; what matters is the general

configuration. First draw lines l and m using the Line Tool (You may have to drag the right

arrow on the Line Straightedge Tool to get the Line Tool rather than the Segment Tool or the

Ray Tool). Next use the Point Tool to place point D inside the acute angle formed by l and m.

Go back to the Line Tool and draw lines n and o passing through D. Then choose the

appropriate pairs of lines and Intersect > Lines to construct points A, B, C, E, and F. If you

have not yet labeled them, select the points A, B, C, D, E, and F in that order and Display >

Label Points, starting at A. Deselect all and then select the lines l, m, n, and o in that order

and Display > Label Lines, starting at l. You can select and drag the line labels so that they

are located as in the diagram.

2. Select A and F and Construct > Segment. Select Display > Color and change the color of the

segment (and subsequent segments) to a contrasting color. With the segment still selected,

o

n

m

l

B

F

A

E

C

D

Select > Midpoint. Similarly, construct segments BE and CD and their midpoints. Label the

midpoint of AF X, the midpoint of BE Y, and the midpoint of CD Z.

3. Select X and Z and Construct > Line. What is the relationship between point Y and this line?

Make a conjecture about the three midpoints and type it into a text box. Drag some of the

points in the original configuration (such as A or F) to see whether your conjecture seems to

hold up. Careful, drag the points slowly and not too far as a small change in the location of a

point can make a big change in the drawing. Save your sketch as <your_name> 4-S-3.

Section 5.1 [32 – 36]

1. Construct 3 points and label them A, B, and C. They will become the vertices of ABC� and

so they must not be on a straight line. You want ABC� to be a scalene triangle (not

equilateral or isosceles). With the points still selected, select Display > Labels and label

them A, B, and C. With the points still selected, select Construct > Segments to create

ABC� .

2. Deselect all. Select the side opposite A, then the side opposite B, and then the side opposite

C, and select Display Labels, labeling the sides a, b, and c in that order. (Make sure you use

lower case for the side labels.) With the sides still selected, select Measure > Lengths.

Deselect all.

3. Measure A BAC∠ = ∠ by selecting B, A, and C in that order and then selecting Measure >

Angle. Deselect all. In a similar fashion, measure B ABC∠ = ∠ and C ACB∠ = ∠ .

4. In a text box, list the names of the sides in order of size from the largest to the smallest. Also,

list the names of the angles in order of size from the largest to the smallest. Use the one-letter

angle name. In the text box, make a conjecture of how one might be able to predict the order

of the angles (largest to smallest) in terms of the order of the sides.

5. Select and drag any vertex of the triangle to change the shape of the triangle. Does your

conjecture seem to hold for all triangles you try? Save your sketch as <your_name> 5-1-5.

Section 7.1 [28 – 35]

In this set of exercises you will construct and explore the Penrose Tiles, named after the physicist

Roger Penrose who discovered them in the 1970s. Some interesting facts about them can be

found on page 262 of the textbook, or see the Wolfram MathWorld article on Penrose Tiles.

The Penrose Tiles you will construct are quadrilaterals called the kite and the dart. They are

formed by starting with a quadrilateral ABED, with AB = BE = ED = DA, and AB ED� ,

DA BE� . As you will see later in the chapter, such a quadrilateral is called a rhombus. This

particular rhombus has DAB 72∠ = ° . Then C is located on the diagonal AE so that AC = AB.

ABCD is the kite and BEDC is the dart. The diagram looks as follows.

1. First, construct the rhombus, as follows. Construct a horizontal line segment AB, about 5 cm

long. (Use the Segment Tool and hold down the shift key to make the line perfectly

horizontal.) Then select A and select Transform > Mark Center. Then select AB and point B

and select Transform > Rotate (72°). Deselect all and select AD and B and Construct >

Parallel Line to construct a line through B parallel to AD. In a similar fashion, construct a

line through D parallel to AB. Deselect all, and select the two lines and Construct >

Intersection to locate E. Hide the straight lines and construct segments BE and ED.

2. Next, construct the kite and dart, as follows. Construct segment AE, a diagonal of ABED,

and give it a contrasting color. Deselect all. Select points A and B in that order and then

Construct > Circle by Center + Point. Leaving the circle selected, select AE and Construct >

Intersection, which is point C. Construct segments BC and DC. Hide the circle. Then ABCD

is the kite and BEDC is the dart. Although the segment AE is not part of the boundary of

either tile, we will refer to it later, so you should not hide it, just leave it a contrasting color.

3. Open a text box, and enter the answer to the question: "Which Penrose Tile is convex?"

4. In the text box, type in the following text, with the blanks replaced by the correct answer.

Since AB ED� , ADE∠ = ___. Since AD BE� , DEB∠ = ___. Since AB ED� , ABE∠ =

___. ABE ADE≅� � by SSS. Therefore DAE BAE∠ = ∠ . The total of the two angles is 72°,

so each is ___. AB = AD = AC because ___ [2 reasons]. Therefore triangles ACE and ACB

are both ___ triangles. Furthermore, ACD ACB≅� � by ___.

5. From what you discovered in problem 4, you should be able to determine the values of the

angles in triangle ABC without measuring. Write these down in the text box, and then verify

all 3 by using Measure > Angle.

6. The diagram consists of two pair of congruent isosceles triangles, and the triangles have 12

angles in all. By either measuring or reasoning, you should be able to find that the measures

of these 12 angles have only 3 different values between them. In the text box, type: "The

measure of the angles in all three triangles all have one of the three values 36°, x°, or y°."

where you replace x and y by the correct numbers.

72°

C

ED

A B

7. Select the Marker Tool and mark BAC∠ by clicking on AB near A and dragging to AC near

A. In a similar manner, mark the remaining 11 triangle angles. Select all angle markers that

identify 36° angles by using the Selection Tool and clicking in the shaded area between the

mark and the vertex of the marked angle, and select Display > Color (Red). Then select all

angle markers that identify x° angles and in a similar way change their color to blue. Then

select all angle markers that identify y° angles and in a similar way change their color to

green. Save your sketch as <your_name> 7-1-7.gsp.

Section 7.3 [45 – 46]

Given a line l and a point P not on the line, it is possible to construct a line through P that is

parallel to l without changing the opening on the compass. In the next exercises, you are guided

to doing this (using Sketchpad rather than a physical compass) and seeing why the method

works.

1. Open a new sketch, and draw a horizontal line, l. Construct a point P above the line, but not

too far; maybe about 5 cm. Construct another line segment, XY, off to the side. This segment

represents the compass opening. It must be somewhat longer than the distance between P and

l; maybe about 8 cm.

2. Select P and then XY, and select Construct > Circle by Center + Radius. Then select line l

and Construct > Intersections. Label the left intersection A. Deselect all.

3. Select A and then XY, and select Construct > Circle by Center + Radius. Then select line l

and Construct > Intersections. Label the right intersection B. Deselect all.

4. Select B and then XY, and select Construct > Circle by Center + Radius. Then select the

circle with center at P and Construct > Intersections. One intersection will be point A. Label

the other intersection C. Construct line PC. It should appear parallel to line l.

5. To show PC really is parallel to l, construct segments AP and BC and give them a contrasting

color. What kind of quadrilateral is ABCP? How do you know? How does this prove that PC

is parallel to l? Type your answer in a text box and save your sketch as

<your_name> 7-3-5.gsp.

Section 8.3 [41 – 48]

1. Start a new sketch, and select Graph > Define Coordinate System. In order to fit all points

required in these exercises, you should move the coordinate system so that the origin is close

to the lower left hand corner of the window. Do this by selecting the origin and dragging it.

2. Plot the point (5,4) by using Graph > Plot Points, entering 5 and 4 in the appropriate fields,

and selecting Plot. Leave the Plot Points dialog box visible and in a similar fashion plot (8,3)

and (3,0) and then click Done. Deselect all, and then select the 3 points in the order created,

and select Display > Label Points starting at A. With the points still selected, select Construct

> Segments to create ABC� .

3. Plot the point (11, 10). Deselect all. Select point A(5,4) and point (11,10) in that order, and

select Transform > Mark Vector. Select points A, B, and C and the segments joining them

and Transform > Translate (Marked Angle). This creates a new triangle. Select just the

vertices of this triangle and select Display > Show Labels. The labels of the new triangle will

be D, B', and C'. Choose only the point labeled D, and then Display > Label Point, and

change its label to A'.

4. Select the point A and A' and Construct > Segment. Then select Display > Line Style and

change segment AA' to a from Medium Thickness to Thin.

5. Plot the points (7,0) and (20,13) and construct the line containing them. Label the line l.

Show that l and AA' are parallel, by showing they have the same slope. Record the common

slope in a text box. Note that Sketchpad will measure the slopes for you if you select the line

and segment and then select Measure > Slopes, but in this case the slopes are very easy to

compute without using Sketchpad. Deselect all.

6. Select line l and then Transform > Mark Mirror. Select all points and segments in A B C′ ′ ′�

and then Transform > Reflect. Select only the vertices of the reflected triangle and select

Display > Show Labels. The new triangle should be A B C′′ ′′ ′′� . The transformation that maps

ABC� to A B C′′ ′′ ′′� is a special type of isometry. Find the name of it on page 313 of your

text, and enter it in your text box.

7. Construct segments AA", BB" and CC". Give them all a color which contrasts with the

previous constructions, and construct their midpoints. Label the midpoints M, N, and P

respectively. Find their coordinates, either by selecting the points and then Measure >

Coordinates, or simply visually. Enter the coordinates in your text box. Also in the text box,

describe how the points are related to line l.

8. In the text box make a general conjecture that says something about the relationship between

points and their images under the type of transformation you identified in Exercise 6. Save

your work as <your_name> 8-3-8.gsp. In the next exercises, you will test that conjecture.

9. Open a new sketch. Do not use a coordinate system. Create and label a scalene triangle ABC

and, off to the side, a segment XY (neither vertical nor horizontal) which will determine the

translation vector. Deselect all.

10. Select points X and Y in that order, and then Transform > Mark Vector. Select all vertices

and sides in ABC� and then Transform > Translate (Marked). Select just all vertices of the

new triangle and Display > Show Labels. They should be A', B', and C'.

11. Select XY and switch to a thin line style for this segment and those following. Use the Point

Tool to create a point (not on any line) that the line of reflection will pass through. With the

point selected, select XY and then Construct > Parallel Line. Name the new line l.

12. Select line l and Transform > Mark Mirror. Then select all vertices and sides of A B C′ ′ ′� and

Transform > Reflect. Select just all vertices of the new triangle and Display > Show Labels.

They should be A", B", and C".

13. Construct lines AA", BB", and CC", change their color, and construct their midpoints. Label

the midpoints M, N, and P in that order. Determine whether the location of these midpoints

matches your conjecture. Drag points A, B, and C to see if the conjecture still holds. Record

your findings is your text box. Save your sketch as <your_name> 8-3-13.gsp.

Chapter 9

In addition to the problems assigned in the textbook, read my Area Notes, and do all problems in

those notes.

Section 10.4 [5 – 6]

1. Construct a scalene triangle ABC that is not too small. (You will be constructing another

triangle inside it, and don't want to have labels overlapping. But you still want to have plenty

of room around the triangle, at least on one side.) Select a point somewhere inside the

triangle and label it D. With the point still selected, select Transform > Mark Center. Now

select the vertices and sides of ABC� and select Transform > Dilate. Select the Ratio for

dilation to be 1/3. A dilation of a figure is always similar to the figure. Deselect all and select

the three vertices of the smaller triangle and Select > Show Labels. The vertices will be

labeled A', B', and C'. Deselect all.

2. Construct line segments AA', BB', and CC' and their midpoints. Label the midpoint of AA'

M, the midpoint of BB' N and the midpoint of CC' P. Select points M, N, and P to create a

new triangle in between the first two triangles.

Triangles ABC and MNP appear similar. In the next exercise, you verify that they are.

3. Label the sides of ABC� as a, b, and c, following the usual convention that a is opposite

vertex A, etc. Label the sides of MNP� as m, n, and o, following the same convention. If

your measurement tools are not already set up to maximum precision, select Edit >

Preferences (Units tab) and set the Precision for all units to "hundred thousandths". Deselect

all, then select a, b, c, m, n, and o and select Measure > Lengths.

4. Select Numbers > Calculate. Select the length of a, then the ÷ key on the calculator, and then

select the length of m. Important: Do not click on the segment itself for the length, which will

do nothing, instead click on the measurement object, such as "a = 12.20862 cm". Resist the

temptation to enter the numbers manually; this will be less accurate and will not change

when you drag points. You should now see the ratio a

mon the screen. In a similar fashion,

compute the ratios b

nand

c

o. The triangles are similar iff these three ratios are the same. You

should verify that they are similar.

5. You have shown that ABC MNO� ∼� . Since the dilation of a triangle is similar to the

original triangle, ABC A B C′ ′ ′� ∼� . Without measuring, you should be able to determine

whether ABC A B C′ ′ ′� ∼� . Open a text box, and type in your conclusion, and the reason(s)

for it.

6. Without changing anything else, drag the dilation center D so that it is outside the triangle

ABC, and (if convenient) so that the entire MNO� is outside triangle. Remember the dilation

ratio is still the same. Notice whether any of the measurements or ratios in Exercise 4 change,

and add your observations to the test box. Save the sketch as <your_name>10-4-5.gsp.

In the next exercises, you will investigate whether the behavior you have observed holds in a

more complicated situation.

7. Select File > Save As and to make a copy of the sketch for you to work on. Choose the name

<your_name>10-4-7.gsp for the sketch. Deselect all, and select segments AA', BB' and CC'.

Delete them using the Delete key on your keyboard. This will also delete any objects that

depend on those segments, such as points M, N, and O and some measurements.

8. Select a point somewhere inside A B C′ ′ ′� and label it E. With E selected, select Transform >

Mark Center. Select the sides and vertices of A B C′ ′ ′� and select Transform > Rotate. To

keep things interesting, do not choose a "special" angle such as 90° or 180°.

9. For the new triangle, select the vertices and Display > Show Labels. The labels should be A",

B", and C". Use Display > Label Points to change these labels to D, E, and F keeping

alphabetical order.

10. Construct segments AD, BE, and CF, and their midpoints. Label the midpoint of AD M, the

midpoint of BE N, and the midpoint of CF O. Construct segments MN, NO, and OM. Label

the sides by selecting segments NO, OM, and MN in that order and Display > Label

Segments starting with m. Hide the sides and vertices of A B C′ ′ ′� .

11. Repeat Exercise 4 for this new sketch. Write up what you have discovered in the text box.

Save the sketch.

Section 10.5 [25 – 30]

1. Open a new sketch, and define a coordinate system. By dragging the origin towards the lower

left hand corner of the window and/or reducing the scale by dragging the unit point towards

the origin, make sure that the graph extends from the origin to at least 20 units to the right

and 15 units up.

2. Plot the following points and Construct > Segments to form ABC� : A(4,8), B(5,3), and

C(10,8). The Line Style should be Medium.

3. Apply the transformation T(a, b) = (2a – 3, 2b – 4), to points A, B, and C, construct the

points, and label them A', B', C'. For example, A A (2 4 3, 2 8 4) A (5,12)′ ′ ′= ⋅ − ⋅ − = so you

should construct a point with coordinates (5,12) and label it A'. Then select A', B', and C' and

Construct > Segments to create A B C′ ′ ′� .

4. Find the fixed point of the transformation T; that is, find the value of a and b such that

T(a, b) = (a, b). To do this, use simple algebra to find the solution of the equations a = 2a – 3

and b = 2b – 4. Open a text box and complete the statement "The transformation has a fixed

point with coordinates ___". Plot the point with these coordinates on your sketch using Graph

> Plot Points, and label it P.

5. Construct the lines AA', BB', and CC'. Choose a Thin line style for the lines. What is the

relationship between these lines and the fixed point P? Put your answer in your text box.

6. Select P and A, and then Measure > Distance. Deselect all, then select P and A' and then

Measure > Distance. Then select Number > Calculate and have Sketchpad compute A P

AP

′.

7. In a similar fashion, compute the ratios B' P

BP,

C P

CP

′and

A

A

′. Enter all ratios in your text box.

8. It appears that T is a dilation. What is the relationship between the center of the dilation and

the fixed point of the transformation? What is the scale factor of the dilation? Type your

answer in your text box.

9. Conjecture a generalization, by completing the following statement and filling in the blanks:

"A transformation of the form T(a, b) = (sa – h, sb – k) is a dilation, with fixed point ___ and

scale factor ___." Careful: The fixed point is not, in general, (h, k). To find the fixed point

you will need to solve the equation a = sa – h for a, and the equation b = sb – h for b. Enter

your conjecture in the text box.

10. Based on your conjecture for Exercise 9, find the values of s, h, and k for a dilation with

center (8, 4) and scale factor of 5, and enter the equation of the transformation in your text

box. Save the sketch as <your_name> 10-5-9.gsp.

Section 11.2 [37 – 41]

1. Open a new sketch and define a coordinate system. Do not change the size of the grid in this

set of exercises, though you may move the origin. Label the origin O. Using Graph > Plot

Points, create the following points and when you have plotted all points label them with the

indicated letters: P(−1,8), Y(1,12), T(5,10), H(3,6), A(9,3), G(6,−3), R(−8,−1), S(−9,7).

2. Using Construct > Segments, draw the quadrilaterals PYTH, HAGO, and ORSP.

3. Select segments OH, OP, and PH and then Measure > Lengths. Using Number > Calculate,

find 2 2

OH PH+ and 2

OP . What can you conclude about OHP� ? Type your conclusion in a

text box.

4. By sight, what special type of quadrilateral do PYTH, HAGO, and ORSP seem to be? Type

your answer in the text box.

5. Assuming your answer to Exercise 4 is true, and using Number > Calculate, what are the

areas of quadrilaterals PYTH, HAGO, and ORSP? Enter your answer in the text box.

6. Deselect all, and then select P, Y, T, and H in that order. Select Construct > Quadrilateral

Interior, and then Measure > Area. In a similar way, let Sketchpad measure the areas

enclosed by HAGO and ORSP. See if these answers are the same as the answers calculated

in Exercise 5. Save your sketch as <your_name> 11.2.6.gsp.

Section 11.3 [48 – 52]

1. Open a new sketch and use the Line Tool to create a horizontal segment. Then mark the left

endpoint of the segment as the transform center, and rotate the right endpoint by 60°. Label

the left endpoint of the segment A, the right endpoint B, and the rotated point as C. Select A,

B, and C and Construct > Segments. ABC� is an equilateral triangle. Let s be the length of

the segment AB.

2. Construct the bisector of ABC∠ . Construct a line through B that is perpendicular to AB.

Label the intersection of the ray and the line as D. Hide the ray and line and construct

segments AD and BD.

3. Construct the bisector of ADB∠ and then construct the intersection of this bisector and AB.

Label the intersection E. Hide the bisector and construct segment DE.

4. Select E and segment AD and Construct > Perpendicular Line. Construct the intersection of

the segment and the perpendicular line and label it F. Hide the perpendicular line and

construct the segment EF. Your drawing should look like the one on page 446 of the

textbook that is above Exercise 48.

5. There are four 30° – 60° right triangles that have labeled vertices. What are they? Put your

answer in a text box.

6. In a 30° – 60° right triangle, the sides of the triangle are in the ratio 3 : 3 : 2 3 (short leg :

long leg : hypotenuse). So, in right triangle ABD, DB 3

AB 3= . Since AB = s, this is equivalent

to 3

DB s3

= . Using a similar argument, you should be able to find that s

EB3

= , so we have

found a complicated way of dividing a line segment into thirds. (What is an easier way?)

Next, find the lengths of the following segments in terms of s: DE, DF, EF, AF. Some

answers involve 3 . Leave them in that form; do not give a decimal approximation. Put

your work in the text box.

7. Using Sketchpad, measure the lengths of AB, EB, DE, DF, EF, AF; and using Number >

Calculate find the ratios EB DE DF EF AF

, , , ,AB AB AB AB AB

and check that they agree with your

answers in Exercise 6. Save your sketch as <your_name> 11-3-7.gsp.

Section 11.4 [48 – 52]

The figure below is drawn on a grid of six squares, each having sides of 1 unit.

1. Find the exact lengths of AB, AC, and BC. Open a new sketch and create a text box. Put your

answers in the text box.

2. Verify that 2 2 2

AB AC BC+ = . Show your work in the text box

3. Find the measure of BAC∠ , then of ABC∠ and ACB∠ .

4. Use Sketchpad to make a copy of the figure (using a rectangular coordinate system), and

verify the results of Exercises 1 and 3.

5. Let ABD∠ = α and EBC∠ = β . By considering right triangles ADB and CEB, find tan(α)

and tan(β) and put the results in your text box. Leave your answers in common fraction form,

not decimal form. You should not need to use Sketchpad to find these answers.

E

A

DB

C

6. A result of trigonometry says that for any angles α and β, tan( ) tan( )

tan( )1 tan( ) tan( )

α + βα + β =

− α β.

Using this formula and the result of Exercise 5, find tan(α+β). Since ABCα + β = ∠ , you can

complete the following equation and enter it into the text box: tan( ABC)∠ = . How does this

agree with the measure of ABC∠ that you got in Exercise 3?

7. Save the sketch as <your_name> 11-4-7.gsp.

Section 11.6 [23 – 35]

1. Open a new sketch, create a coordinate system, and plot the following points A(−8,−5),

B(−1.4), C(10,7), D(3,−2). Select points A, B, C, and D in that order and Construct >

Segments to construct a quadrilateral. Also construct the diagonals AC and BD.

2. Measure the slopes of the lines containing the six segments you constructed in Exercise 1.

3. Copy the following into your text box, filling in the blanks. "Two lines that have the same

slope are parallel to one another. Therefore __ __� , __ __� , and quadrilateral ABCD must

be a __."

4. Measure the lengths of the sides of ABCD� . Open a text box and describe what type of

quadrilateral ABCD is. You should be able to give a more detailed answer than in Exercise 3.

5. What is it about the slopes of AC and BD that tells you the diagonals are perpendicular to

each other? Put your answer in the text box, and save your sketch as


Chapter 11 Summary & Review [48 – 50]

1. Open a new sketch, and a text box. Type in the answer to Exercise 48, page 479 of your

textbook.

2. In right hand drawing above Exercise 48, note that ADE� is isosceles and

360DEA 120

3

°∠ = = ° . What is the measure of ADE∠ and DAE∠ ? Put your answer in the

text box.

3. What is the measure of BCF∠ and CBF∠ ?

4. Create a sketch similar to the right hand drawing above Exercise 48 at a scale of 10 cm

represents 1 mile. To do this, start a new sketch and define a coordinate system. Plot points

A(0,0), B(10,0), C(10,10) and D(0,10). Select the points in order and construct segments.

Then continue with Exercise 5.

5. Rotate segment AD 30° around center D and −30° around center A, and construct the

intersection of the two rotated segments to locate E. In a similar fashion, rotate BC around

centers B and C and intersect the rotated segments to locate F. Construct segments AE and

EF.

6. In the text box, describe how you know that AE = DE = EF = BF. This means that the sum of

the lengths of the 5 segments in the diagram is equal to EF 4 AE+ ⋅ . Measure the lengths of

EF and AE, and calculate EF 4 AE+ ⋅ . Divide the result by 10 (just move the decimal point)

to get the number of miles of road needed in the second solution.

7. Which solution is better, the 3-segment solution or the 5-segment solution? Put the answer in

your text box and save your sketch as <your_name> 11-S-6.gsp.

Section 12.1 [43 – 47]

You will be drawing a number of circles, each with the same "unit" radius. It's convenient to

have the radius be a small integer number of centimeters, where you might want to change the

integer to make the whole diagram a reasonable size. To do this, it is convenient to use a

parameter, which will be introduced below.

1. Select Number > New Parameter. Set Name = r and Value = 4, and click OK. Construct a

point near the middle of the window and label it P. Select P and the parameter r and then

Construct > Circle by Center + Radius.

2. Construct three points on the circle, that divide the circle into three roughly equal parts, and

label them X, Y, and Z. Open a text box and type "Since X, Y and Z all lie on the circle with

center P and radius 4 cm, __ = __ =__ = 4 cm", where you have filled in the blanks. Now

construct segments PX, PY, and PZ and use Measure > Length. Describe what you find in

your text box.

3. Construct three new circles with centers X, Y, and Z, and radius 4 cm. In your text box,

explain why all three circles pass through P. Hide the circle with center P. We will refer to

the three circles by their center.

4. Select circles X and Y and Construct > Intersections. One intersection will be P. Label the

other A. Similarly, label the intersection of circles Y and Z (not P) B, and the intersection of

circles Z and X (not P) C. Your drawing should look like the one on page 489 of the

textbook, bottom of the second column.

5. Draw XA, XC, YA, YB, ZC and ZB. See if you can tell what the lengths of these segments

are, and then measure them to confirm your idea.

6. You have created 3 quadrilaterals that look like three faces of a cube. Finish constructing the

"cube" by constructing a line through A, parallel to YB and then a line through B parallel to

AY. Find the intersection of these two lines, and label it O. Which circle does it appear to be

on? Record your answer. Hide the lines and construct segments OA, OB, and OC. Measure

their lengths.

7. You have drawn six quadrilaterals, corresponding to the six sides of a cube. How many of

these six quadrilaterals are rhombi? Put your answer in the text box.

Section 12.2 [19 – 23]

For an introduction to this set of exercises, see the beginning of page 494 in your textbook. The

same kind of problem occurs in mechanical engineering, where the corner of a sharp object is

rounded to make it less likely that a person could be injured handling it. In engineering, this is

called a fillet.

1. Open a new sketch and construct and label lines l and m, and their intersection P, that look

similar to the diagram above Exercise 19 on page 494.

2. Draw a circle centered at P with any convenient radius. The smaller the radius, the sharper

the curved corner will be. Find the intersections of the circle with l and m and label two of

the intersections A and B as in the diagram.

3. Select A and l and Construct > Perpendicular line. Deselect and select B and m and

Construct > Perpendicular line. With both perpendicular lines selected, select

Construct > Intersection, and mark the Intersection O.

4. Construct a circle with center O passing through point A. You will prove in the following

exercise that this circle passes through B as well, and is tangent to l at A and to m at B.

5. You want to "erase" most of the circle so that only the short arc between A and B remains, as

in the textbook diagram. To do this, first use the point tool to create a point on the circle

between A and B, and label it C.

6. Deselect all, then select the points A, C, and B in that order and select

Construct > Arc through 3 Points. Deselect all, then click on the part of the circle that you

want to erase and select Display > Hide Circle. This will leave only the arc visible.

7. Construct segment OP, an auxiliary segment, and color it red. Open a text box, and answer

Exercises 20 – 23 on page 494 of the textbook in the text box. Save your sketch as


Section 12.6 [18 – 21]

In these exercises, you will construct the tangents to a circle from an external point, and will

explain why the method works.

1. Construct a circle with the circle tool, and an external point with the point tool. Label the

center of the circle O and the external point P.

2. Construct segment OP, and then construct the midpoint of OP and label it M.

3. Construct a circle with center M passing through O (and P). Construct the intersections of the

two circles, and label them A and B. Construct rays PA and PB. These are the desired

tangents.

4. Construct the auxiliary segments PA and PB. What kind of angles are OAP∠ and OAP∠ and

how do you know? Put your answers to this and the following questions in a text box.

5. What relation does PA have to the radius OA? What relation does PB have to radius OB?

6. How do you know that PA and PB are tangent to the circle O? Save your sketch as



A cat is standing on the middle rung of a ladder which is leaning against a wall. The bottom of

the ladder starts to slide away from the wall, while the top stays along the wall. To find out the

path that the cat takes, open the handout Cat On A Ladder.gsp, and follow the instructions. Type

your answers into a text box in the sketch and save as <your_name> Cat On A Ladder.gsp.

Section 13.3 [27 – 30]

1. Start a new sketch. Create ABC� . Measure the lengths of the three sides and drag vertices

until the lengths of the sides are approximately 8 cm, 9 cm, and 10 cm. (The exact values are

not important, however make sure that the triangle does not have two equal sides.)

2. Create the circumcircle of ABC� as follows. Locate the midpoint of AB, and construct the

line passing through the midpoint and perpendicular to AB. Then locate the midpoint of BC,

and construct the line passing through the midpoint and perpendicular to BC. Construct the

intersection of the two lines, and label it O. Construct the circle with center O and passing

through A. This circle will pass through points A, B, and C and thus is the circumcircle of

ABC� .

3. Create the incircle of ABC� as follows. Construct the bisector of BAC∠ . Construct the

bisector of ABC∠ . Construct the intersection of the two rays you just constructed, and label

it I. Hide the lines. Construct a line through I that is perpendicular to AB. Construct the

intersection of this line and AB. Draw the circle with center I that passes through the point

you just constructed. Then hide the line.

4. Use the point tool to create a point somewhere on the circumcircle, and label it D.

5. Construct the rays from the point D that are tangent to the incircle I as follows. Construct

segment DI. Construct M, the midpoint of DI. Construct the circle with center M that passes

through I. Construct the intersections of this circle and circle I, call them S and T. Construct

the rays DS and DT. These are the tangent lines. (See the GSP exercises for section 12.6 for

why this works.)

6. Construct the intersection of DS and circle O and the intersection of DT and circle O. Label

these points E and F, in either order. Hide the rays DS and DT, and construct segments DE,

EF, and FD. Measure the lengths of these three segments.

7. It should appear that EF is tangent to circle I. To verify that this is so, select circle I and

segment EF, and Construct > Intersections. Notice that there is only one intersection point,

which shows that EF is tangent to circle I.

8. To simplify the diagram, hide everything except triangles ABC and DEF, Circles I and O,

and the measurements. By looking at the measurements of the lengths of the sides of the two

triangles, you should be able to determine whether or not they are congruent. In your text

box, complete the following sentence: "It appears __ for two triangles that are not congruent

to have the same incircle and circumcircle." where you replace the blank with the word

"possible" or "impossible".

9. Drag point D slowly around the circumcircle, watching to see if EF remains tangent to the

incircle and also watching what happens to the measurements of the sides of DEF� (do they

change?). Describe what you observed in the text box, and save the sketch as


You have investigated the simplest case of a remarkable theorem called Poncelet's Porism. If you

are curious, you can find out exactly what this says by looking up Poncelet's Porism in Wolfram

MathWorld online.

Section 13.4 [27 – 31]

On page 352 of the textbook, the altitude of a triangle was defined to be "A perpendicular line

segment from a vertex of a triangle to the line of the opposite side". The reason for the words

"the line of" is shown in Exercises 1 – 4 below.

1. Start a new sketch and create ABC� where C is an obtuse angle. Select segment AC and

point B and Construct > Perpendicular line. Next select points A and C and Construct > Line.

Select the two lines you have created and construct their intersection. Label it D. The altitude

is segment AD.

2. Construct the other two altitudes of ABC� and label them BE and CF.

3. Are the three line segments AD, BE, and CF concurrent? Explain, putting your answer in a

text box.

4. Are the three lines AD, BE, and CF concurrent? If so, the point of concurrency is called the

orthocenter. Put your answer in a text box.

5. Which of the following points can lie outside a triangle: its incenter, its orthocenter, its

centroid, its circumcenter? Hint: If you are not sure, use Sketchpad to create an obtuse

triangle and experiment. Put your answer in a text box and save the sketch as <your_name>

14-4-5.gsp.

extra problems for jacobs using gsp

Documents

point c

point tool

text box

new point

graph plot points

display label points

points d

text tool