October 31, 2013

1.Similar Triangles:• Corresponding ANGLES are congruent• Corresponding SIDES are proportional• Have the same shape, but not the same size

1

2

4 3

6

12

4-1 Triangle Similarity with Dilations

2. Dilation:• a transformation that enlarges or reduces a

pre-image to create a similar image

Center of Dilation

A

B

C

A`

B`

C`

D

A dilation requires a center point and a scale factor. The

letter r usually represents the scale factor.

In the above figure - Triangle A'B'C' is a dilation of triangle

ABC

3.Scale Factor:• Is the ratio: • the distance from the center of dilation to a point on the

image: to the distance from the center of dilation to the corresponding point on the pre-image.

• When |r| is greater than 1, the dilation is an enlargement.• When |r| is between 0 and 1, the dilation is a reduction. • If r>0, P' lies on CP, and CP' = r(CP)• If r<0, P' lies on CP' (the ray opposite CP) and |r|(CP)

4. Dilations preserve angle measure, betweenness of points, and collinearity, but do NOT preserve distance. Therefore, a dilation is a similarity transformation.

October 31, 2013

Vocabulary

Image:

Preimage:

Dilation:

Center of Dilation:

Scale Factor:

Similar:

ex) Find the measure of the dilation image or the preimage using the given scale factor.

a) AB = 12, r = 2 b) A'B' = 36, r = 1/4

5. Constructing dilations.ex) Draw the dilation image of triangle JKL with center C and r = -1/2

K

LJ

C

Steps:

1) Draw CJ, CK, and CL. Since r is negative, J', K', and L' will lie on CJ', CK' and CL' respectively.

2) Locate J', K', and L' so that CJ' = (1/2)(CJ), CK' = (1/2)(CK), and CL' = (1/2)(CL).

3) Draw triangle J'K'L'

5. Triangle J'K'L' is a dilation of Triangle JKL. The center of the dilation is the origin.

a. List the coordinates of the vertices of Triangle JKL and Triangle J'K'L'. How do the coordinates of the image compare to the coordinates of the pre-image?

b. What is the scale factor?

c. How do you think you can use the scale factor to determine the coordinates of the vertices of an image?

*The point (x,y) dilated can be described as (kx,ky) when the center of dilation is at the origin.

Secondary*Math*2* IN1CLASS* * * * * * * * *Triangle*Similarity*with*Dilations***** * * * * * **

1. Use*quadrilateral*ABCD*shown*on*the*grid*to*complete*part*(a)*through*(c).*a. On*the*grid,*draw*the*image*of*quadrilateral*ABCD*

dilated*using*a*scale*factor*of*3*with*the*center*of*dilation*at*the*origin.**Label*the*image*JKLM.*

b. On*the*grid*draw*the*image*of*quadrilateral*ABCD*dilated*using*a*scale*factor*of*0.5*with*the*center*of*dilation*at*the*origin.**Label*the*image*WXYZ.*

c. Identify*the*coordinates*of*the*vertices*of*quadrilaterals*JKLM*and*WXYZ.*********

**

2. Determine*the*scale*factor**of*the*following:* *a) Scale*factor:*_______ * * * *

* ** ** ** ** **

** **

3. ABCΔ *has*vertices*A*(1,2),*B*(3,6),*and*C*(9,7).*What*are*the*vertices*of*the*image*after*a*dilation*with*a*scale*factor*of*4,*using*the*origin*as*the*center*of*dilation?*******

4. GHIΔ *has*vertices*G*(0,20),*H*(16,24),*and*I*(12,12).*What*are*the*vertices*of*the*image*after*a*

dilation*with*a*scale*factor*of* 34*using*the*origin*as*the*center*of*dilation?*

*****

**The*following*polygons*are*similar.**Find*x*and*y.**5/6.** * * * * * * 7/8.*

**************9.* ABCΔ *has*vertices*A*(1,2),*B*(3,6),*and*C*(9,7).*What*are*the*vertices*of*the*image*after*a*dilation*with*a*scale*factor*of*1/4,*using*the*origin*as*the*center*of*dilation?********

10. GHIΔ *has*vertices*G*(0,20),*H*(16,24),*and*I*(12,12).*What*are*the*vertices*of*the*image*after*a*dilation*with*a*scale*factor*of* 5 *using*the*origin*as*the*center*of*dilation?*

*

October 31, 2013

As you recall from yesterday:

Identify all of the corresponding congruent angles and all of the corresponding proportional sides using the similar triangles shown.

R S

T

W X

Y

In two similar figures all corresponding angles are congruent and corresponding sides are proportional.

4-2 Triangle Similarity Theorems Construct triangle D'E'F' using only D and E in triangle DEF as shown. Make all corresponding lengths of triangle D'E'F' different from the side lengths of triangle DEF.

D

EF

Measure the angles and sides of triangle D'E'F' and triangle DEF . Are the two triangles similar? Explain your reasoning .

D

EF

In triangles DEF and D'E'F', two pairs of corresponding angles are congruent . Determine if this is sufficient information to conclude that the triangles are similar .

Angle-Angle Similarity Theorem:If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar .

A

B

C

D

E

F

October 31, 2013

Explain why this similarity theorem is Angle-Angle instead of Angle-Angle-Angle .

Construct triangle D'E'F' by doubling the lengths of sides DE and EF . Construct the new D'E' and E'F' separately and then construct the triangle. This will ensure a ratio of 2:1. Do not duplicate angles.

D

E

F

D

E

F

Measure the angles and sides of triangle D'E'F' and triangle DEF. Are the two triangles similar? Explain your reasoning.

Two pairs of corresponding sides are proportional. Determine if this is sufficient information to conclude that the triangles are similar.

D

E

F

Measure the angles and sides of triangle D'E'F' and triangle DEF. Are the two triangles similar? Explain your reasoning.

Two pairs of corresponding sides are proportional. Determine if this is sufficient information to conclude that the triangles are similar.

October 31, 2013

Construct triangle D'E'F' by doubling the lengths of sides DE , EF, and FD . Construct the new side lengths separately, and then construct the triangle. Do not duplicate angles.

D

E

F

Measure the angles and sides of triangle D'E'F' and triangle DEF. Are the two triangles similar? Explain your reasoning.

D

E

F

Three pairs of corresponding sides are proportional. Determine if this is sufficient information to conclude that the triangles are similar.

Side-Side-Side Similarity Theorem:

If all three corresponding sides of two triangles are proportional, then the triangles are similar .

A

B

C

D

E

F

Determine whether UVW is similar to XYZ . If so, use symbols to write a similarity statement .

24 meters

33 meters

36 meters

22 meters

24 meters

16 meters

W

X

Y

Z

U

V

October 31, 2013

Construct triangle D'E'F' by duplicating an angle and doubling the length of the two sides that make up that angle. Construct the new side lengths separately, and then construct the triangle.

D

E

F

Measure the angles and sides of triangle D'E'F' and triangle DEF. Are the two triangles similar? Explain your reasoning.

D

E

F

Two pairs of corresponding sides are proportional and the corresponding included angles are congruent. Determine if this is sufficient information to conclude that the triangles are similar.

Side-Angle-Side Similarity Theorem:If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the trianglesare similar .

A

B

C

D

E

F

W

XZ

H P

Explain

If yes write a similarity statement

Secondary*Math*2* INCLASS* * * * * * * *452*Triangle*Similarity*Theorems* * * * * * * ***Name*a*theorem*that*can*be*used*to*prove*that*the*two*triangles*are*similar.**Then,*write*a*similarity*statement.*If*you*are*using*any*proportional*sides*be*sure*to*show*that*they*are*proportional.**! 1.** * * * 2.* * * * 3.*!

**!!

!

!

**Determine*which*two*of*the*three*given*triangles*are*similar.**Find*the*scale*factor*for*the*pair.!!

* 4/5.********

***

Determine*whether*the*triangles*can*be*proven*Similar.*If*they*are*similar*name*the*theorem.*If*they*aren’t*similar*explain*why.*If*you*are*using*any*proportional*sides*be*sure*to*show*that*they*are*proportional*or*not*proportional.**

6.** * * * 7.* * * * 8.***********

**

October 31, 2013

A

BC

D

E

Given:

Prove: Solve the following proportions for x:

4-3 Triangle Proportionality

Angle Bisector/Proportional Side Theorem: “A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle.”

On the map, North Craig Street bisects the angle formed between Bellefield Avenue and Ellsworth Avenue.

• The distance from the ATM to the Coffee Shop is 300 feet, the Coffee Shop to the Library is 500 feet, and from your apartment to the Library is 1200 feet.

Determine the distance from your apartment to the ATM.

October 31, 2013

bisects What is the measure of ?Practice

Triangle Proportionality Theorem: “If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. ”

Converse of the Triangle Proportionality Theorem: “If a line divides two sides of a triangle proportionally, then it is parallel to the third side.”

If then =

Proportional Segments Theorem: “If three parallel lines intersect two transversals, then they divide the transversals proportionally."

October 31, 2013

Using the Triangle Proportionality Theorem and triangle ACD, what can you conclude?

Using the Triangle Proportionality Theorem and triangle FDC, what can you conclude?

Through any 2 pts there is exactly 1 line. Draw to form

Label the point where intersects , H.

What property of equality will justify the prove statement?

Triangle Midsegment Theorem: “The midsegment of a triangle is parallel to the third side of the triangle and is half the measure of the third side of the triangle.”

Carson told Alicia that using the Triangle Midsegment Theorem, he could conclude that is a midsegment. Is Carson correct? How should Alicia respond if Carson is incorrect?

Ms. Zoid asked her students to determine whether

is the midsegment of , given TY = 14cm and RD = 7cm.

Alicia told Carson that using the Triangle Midsegment Theorem, she could conclude that is a midsegment. Is Alicia correct? How should Carson respond if Alicia is incorrect?

Ms. Zoid asked her students to determine whether

is the midsegment of , given

October 31, 2013

The truss for a barn roof is shown below. bisects and bisects . is an equilateral triangle. Calculate the perimeter of the truss.

Solve for x.Bridge Over Canyon:

A bridge is needed to cross over a canyon. The dotted line segment connecting points S and R represents the bridge. The distance from point P to point S is 45 yards. The distance from point Q to point S is 130 feet. How long is the bridge?

Cut out the three triangles and arrange the triangles so they have same orientation. Are any of the triangles similar to each other? If so, which triangles are similar? Justify your response.

4-4 Geometric Mean

October 31, 2013

Complete the following similarity statements using your triangles:

Write the corresponding sides of as proportions:

Write the corresponding sides of as proportions:

Write the corresponding sides of as proportions:

Right Triangle Altitude Similarity Theorem:

The altitude to the hypotenuse of a right triangle forms two triangles ___________ to the original right triangle.

Are there any geometric means found in the proportions on the previous page?

Geometric Mean: The geometric mean between two positive numbers a and b is the positive number x such that

P

Q

RS

Write the 3 geometric mean proportions that are created when the altitude is drawn to the hypotenuse of a right triangle.

Secondary Math 2 – INCLASS NAME_______________________ 4-3 Triangle Proportionality PERIOD ____________________ 1. Complete. a. AD = 21, DC = 14, BC = 8, AB = ___ b. DC = 18, AC = 40, AB = 25, AD = ___

c. AB = 27, BC = x, CD = 43x, AD = x, AC = ___

2. In the figure, D is the midpoint of AB and E is the midpoint of BC . If AC = 18, what is DE?

3. In the figure, NQ OP and NQ = 4, OP = 6 and MQ = 8. If NO = 2, how long is QP? 4. In a triangle, the lengths of the sides are 3, 7, and 8. If the perimeter of a similar triangle is 54, what is the length of the longest side of the larger triangle?

D

A B C

B

D E

C A

Q P M

N

O

5. Write all of the equal proportions for the figure. f g

e

d

October 31, 2013

Find the length of the bridge needed to cross the canyon.

P

Q

RS45

130

x

Find the geometric mean between 20 and 5.

Find the geometric mean between 24 and 4.

The geometric mean between two numbers is 20. One of the numbers is 50, find the other number.

Solve for x:

October 31, 2013

You are standing 15 feet from a tree. Your line of sight to the top of the tree and to the bottom of the tree forms a 90-degree angle as shown in the diagram. The distance between your line of sight and the ground is 5 feet. Estimate the height of the tree.

Solve for x, y, and z.

Solve for x, y, and z.

4-4 IN-Class

Right Triangle Proportions

ACBR is a right angle and CN AB⊥

1. If m1= k , then mA = _____ , m2 = _____ , and mB = _____ .

2. Since , AB ACACB ANCAC

Δ Δ =:

3. The diagram shows a right triangle with the altitude drawn to the hypotenuse.

a. p is the geometric mean between ______ and ______.

b. e is the geometric mean between ______ and ______.

c. f is the geometric mean between ______ and ______.

Find the geometric mean between the two numbers.

4. 1 and 50 5. 6 and 10 6. 2 and 110

7. Marsha wants to walk from the parking lot through the forest to the clearing, as shown in the diagram. She knows that the forest ranger station is 154 feet from the flag pole and the flag pole is 350 feet from the clearing. How far is the parking lot from the clearing?

!

Each diagram shows a right Δ with the altitude drawn to the hypotenuse. Find the values of the variables.

8. 9. 10.

e!

j!

p!

k!

f!

m!

A! B!N!

C!

2!1!

y!x! z!

8! 8!

9!

x!

y!z!

15!

z!

2!

y!

6!

x!