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1
Geometry Green Problem Examples
Point, line, line segment, ray, intersecting lines, parallel lines Overview of angles (not assessed) Polygons Triangle classification – isosceles, equilateral and scalene Classifying Quadrilaterals Perimeter and area • Derive formulae for square, rectangle, parallelogram and triangle • Composite area
Volume • Solid figures • Volume of a prism
Blue – Additional Concepts Must be proficient at the above in addition to:
Perpendicular lines Vertical angles Interior angles of triangle Classifying angles (assessed) Circumference and area of a circle Complex word problems on area, perimeter and volume
Black – Additional Concepts Must be proficient at the above green and blue level in addition to:
Exterior angles of triangle Angles in a quadrilateral Pythagorean Theorem and associated word problems.
Black will be permitted to use a calculator during the test when working on this and can use the pi button.
2
Geometry – Green Problems
Types of Angles
Example Name the following angles.
a.
b. c.
d.
e.
Solution a.
Right angle
b.
Acute angle
c.
Obtuse angle
d.
Reflex angle
e.
Straight angle
Classify each angle as acute, obtuse, right, or straight.
1.
2.
3.
4.
5.
6.
7. 84c 8. 179c 9. 90c
10. 180c 11. 12c 12. 91c
Tell which kind of angle is formed when each type of angle is bisected. Make a model to verify your answer to each exercise.
13. a right angle 14. an acute angle
15. a straight angle 16. an obtuse angle
3
Understanding Polygons
1. Give one reason why the following are not polygons:
2. Which of the following are regular polygons:
[Note: Angles marked with the same symbol are equal in size.]
3. Using the following code, name the polygons which follow: tri-3, quad-4, penta-5, hexa-6, hepta-7, octa-8, nona-9, deca-10, dodeca-12 a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
4. Draw an example of a. a quadrilateral b. an equilateral
triangle c. a hexagon
d. a decagon e. a regular pentagon f. an octagon
5. Draw and name polygons with the following descriptions: a. six equal sides and six equal angles b. three equal sides c. five equal sides, but with unequal angles.
4
6. Classify the following shapes as regular (R) or irregular (I) polygons: a.
b.
c.
d.
e.
f.
Test Yourself on Polygons Tell whether each polygon is a regular polygon. If not, tell why.
7.
8.
9.
10.
11.
12.
Sketch each figure. Find the measure of an exterior angle. Then find the measure of an angle of the polygon. 13. regular hexagon 14. regular octagon 15. regular nonagon
16. Which type of quadrilateral is an equiangular quadrilateral, but is not an equilateral
quadrilateral?
17. Which type of quadrilateral is regular?
18. Which type(s) of regular polygon has each exterior angle equal to each interior angle?
19. Which type(s) of polygon has at least two exterior angles which are obtuse angles?
20. Which type of quadrilateral is equilateral but not equiangular?
5
Areas Area of a rectangle Area = Length x width Area = 4 x 6 Area = 24 cm2 Area of a Parallelogram Area = base x height Area = 8 x 5 (7 is not the height, it’s the length of a side) Area = 40 cm2 Area of a triangle Area = 2
1 x base x height
Area = 21 x 7 x 4
Area = 14 cm2
Perimeter and Area
Find the area.
1.
2.
3.
4.
5. 6.
6
Change all units to cm.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
7
Calculate the area of each figure in Problems 19 – 23. Remember to include the unit in each answer. 19. parallelogram
Area __________
20. rectangle
Area __________
21. parallelogram
Area __________
22. triangle
Area __________
23. triangle
Area __________
Area of a Composite Figure A figure obtained from combining two or more different figures is called a composite figure and its area, is found by adding together the areas of the individual figures. Find the area of the triangle. 24. 14 m
3 m14 m
3 m14 m
3 m
25.
26.
Find the area of the figure. 27. 6 ft 2 ft
3 ft
6 ft 2 ft
3 ft
6 ft 2 ft
3 ft
28.
29.
Find the missing dimension of the triangle described. 30. area: 27 in 2
base: 9 in.
31. area: 64 cm2 height: 8 cm
32. area: 144 ft2 base: 12 ft
8
Find the area of the figure when the given lengths, in inches, are a = 6, b = 8, and c = 7. 33.
34.
In Exercises 35 – 36, use the following information. The area of a wing of an airplane can be approximated by finding the area of a triangle.
35. Approximate the area of the wing shown.
36. What is the approximate area of both of the wings?
9
Geometry – Green Review Problems Find the area of these figures.
1.
2.
3.
4. radius = 0.5 cm
5.
6.
7. Which figure is a polygon? a.
b.
c.
d.
8. Which figure is not a polygon? a.
b.
c.
d.
9. Classify a polygon having seven sides. a. hexagon b. heptagon c. septagon d. decagon
10. Classify triangle X by its sides and by its angles. a. scalene, obtuse b. scalene, right c. isosceles, right d. scalene acute
10
11. Classify triangle Y by its sides and by its angles. a. isosceles, right b. isosceles, obtuse c. isosceles, acute d. equilateral, acute
12. Classify triangle Z by its sides and by its angles. a. equilateral, acute b. isosceles, obtuse c. equilateral, right d. equilateral, obtuse
13. The measure of an angle is 45c . Classify the angle. a. acute b. straight c. obtuse d. right
14. The measure of an angle is 180c . Classify the angle. a. acute b. straight c. obtuse d. right
15. The measure of an agle is 135c . Classify the angle. a. acute b. straight c. obtuse d. right
16. A rectangular house is 15 m long. If the area of the house is 345 m2, how wide is the house? (Hint: draw a picture to represent the measurements).
17. Challenge Problem. A swimming pool is 40 ft. by 20 ft. A fence surrounding the pool is 40 ft. by 70 ft. How much deck space is around the pool? (Hint: Draw a picture to represent the measurements).
18. Calculate the area of this square:
19. a. Add a dashed line to divide this shape into two rectangles.
b. Calculate the area of the whole shape.
11
20. This is a parallelogram: a. Draw in the two diagonals. b. True or false?
i. the diagonals are the same length ii. the diagonals bisect each other (cut each other in half) iii. the diagonals meet at right-angles
21. Here are some quadrilaterals:
isosceles trapezium rhombus kite Write down the names of the quadrilaterals that have: a. one pair only of opposite sides the same length b. no pairs of parallel sides c. diagonals that are the same length d. diagonals that meet at right-angles
Calculate the perimeter of these shapes: 22.
5 cm
5 cm
3 cm3 cm
5 cm
5 cm
3 cm3 cm
23.
10 cm8 cm
6 cm
10 cm8 cm
6 cm
24. 11 m
12 m13 m
16 m
11 m
12 m13 m
16 m
25.
18 cm
20 cm
8 cm18 cm
20 cm
8 cm
12
Calculate the area of each parallelogram: 26.
27.
28.
29. A cabbage patch is rectangular in shape, and measures 40 m by 36 m. Avocado trees will be planted on all four boundaries, at least 11 m apart. a. Calculate the perimeter of the patch. b. What is the largest number of avocado trees that can be planted around the
outside?
30. a. Draw two equilateral triangles (ABC) and (BCD) that share one side (BC) in common.
b. What kind of quadrilateral is ABCD? Explain.
13
Notes for Blue and Black
Title and Symbol Theorem Figure Angles at a point
The sum of the sizes of the angles at a point is 360c .
i.e., a + b + c = 360
Adjacent angles on a
straight line
The sum of the sizes of the angles on a line is 180c (the angles are supplementary).
i.e., a + b = 180
Adjacent angles in
right-angle
The sum of the sizes of the angles in a right-angle is 90c (the angles are complementary).
i.e., a + b = 90
Vertically opposite angles
Vertically opposite angles are equal in size.
i.e., a = b
Corresponding angles
When two parallel lines are cut by a third line, then angles in corresponding positions are equal in size.
i.e., a = b
Figure Term Meaning
Scalene triangle
triangle with no equal sides
Isosceles triangle
triangle with at least two equal sides
Equilateral triangle
Triangle with three equal sides
14
Title and Symbol Theorem Figure Alternate angles
When two parallel lines are cut by a third line, then angles in alternate positions are equal in size.
i.e., a = b
Co-interior angles
When two parallel lines are cut by a third line, co-interior angles are supplementary.
i.e., a + b = 180
Angles of a triangle
The sum of the interior angles of a triangle is 180c .
i.e., a + b + c = 180
Exterior angle of a triangle
The size of the exterior angle of a triangle is equal to the sum of the interior opposite angles.
i.e., c = a + b
Figure Term Meaning
Right angle
Angle = 90c
Straight angle
Angle =180c
A revolution
Angle = 360c
Acute angle
0c < angle < 90c
Obtuse angle
90c < angle < 180c
Reflex angle
180c < angle < 360c
Complementary angles
add up to 90c
a + b = 90
Supplementary angles
add up to 180c
a + b = 180
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Area
Figure Formula Example Square
A = s 2
S = side
A = s A = 3 A = 9 in 2
Rectangle
A = l x w l = length w = width
A = l x w A = 5 x 3 A = 15 square m.
Parallelogram
A = b x h
b = base h = height
A = b x h A = 8 x 2 A = 16 square feet.
Triangle
A = 21 bh
b = base h = height
A = 21 bh
A = 21 x 8 x 6
A = 24 cm 2
Circle
A = rr 2 r = radius r = 7
22 or 3.14
A = rr 2
A = 7122 x 7 1 x 7
A = 154 m2 A = rr 2 A = 3.14 x 2 x 2 A = 8.28 ft2 (d = 4 so r = 2)
Trapezoid
A = 2a + b
c mh
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Simple Equations and Geometry Sometimes the relationships between angles can be expressed using algebra. Example The diagram shows a
triangle. The second angle is twice as big as the first. The third angle is three times as big as the first. What are the values of each angle?
Answer We use the fact that the angles of a triangle add to
180c x + 2x + 3x = 180c (E sum of D)
6x = 180c x = 6
180
x = 30c The other two angles must be 2 x 30c = 60c and 3 x 30c = 90c
Although the value of x is unknown, it is always the same value in any particular diagram.
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Quadrilaterals
Example 1 Find x, giving a brief reason for your answer.
The figure is a parallelogram. ` 3x = x + 100 {opposite angles of a parallelogram} ` 2x = 100 ` x = 50
Example 2 Prove that “the opposite angles of a parallelogram are equal in size.”
To prove: that given parallelogram ABCD then x = y Prove: Construct diagonal BD. Let EABD be p, ECBD be q, and EBDA be r, EBDC be s.
Now in DABD x + p + r = 180, {Angles in a triangle} and in DCBD y + s + q = 180. {Similarly} But p = s {Alternate angles are equal} and r = q {Similarly} ` x = y So, opposite angles of a parallelogram are equal in size.
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Special Triangles In this section we will look at the special properties of isosceles and equilateral triangles. Congruence can be used to prove these properties. In an isosceles triangle • the base angles are equal in size, • the line joining the apex to the midpoint of
the base is perpendicular to the base, and • this same line also bisects the angle at the
apex. Note: The converse of these statements is also true. In particular: If a triangle has two equal angles then the triangle is isosceles. In an equilateral triangle: • all angles are equal and must each be 60c • a line drawn from any vertex to the
midpoint of the opposite side (called a median)
• the medians are concurrent (i.e., meet at one point).
Note: • If you need to construct a 60c angle just construct an
equilateral triangle of any size. • An equilateral triangle is a special type of isosceles triangle. Example: Find the value of x, giving brief reasons:
Since AC = BC DABC is isosceles and ` EBAC = EABC = 52c ` x = 52 + 52 {Exterior angle of a triangle} ` x = 104
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Example 1: Find the value of a, giving a brief reason for your answer.
90 + a + 40 = 180 {Angles on a line} ` a + 130 = 180
`a = 50
Example 2: Find the value of the pronumeral in each triangle, giving a brief reason for each answer: a. b.
a. 2x + x + (x + 20) = 180 {angles in a triangle} ` 4x + 20 = 180 ` 4x = 160 ` x = 40 b. Angle BCE measures 60c . {Angles on a line} Now 60 + x = 140 {Exterior angle of a triangle} ` x = 80
Example 3: Use angle theorems to find the value of the pronumeral. Give a brief reason for your answer. 2x - 100 = x {corresponding angles on parallel lines are equal} ` 2x - x = 100 ` x = 100
Example 4: Find a given:
Extend DC to meet AB at X. Now EBXD and EXDE are equal alternate angles on parallel lines
` EBXD = 60c Thus a = 60 + 40 {‘exterior angle of triangle’ theorem} i.e., a = 100
20
Parts of a Circle
centre circumference radius diameter
chord arc sector segment
tangent area inside semicircle a circle
21
Geometry – Blue Problems
1. In each diagram, the angle marked x is 70c . Give a reason why. Use one of the four reasons given above. a.
b.
c.
d.
2. In a test on angles, Lee wrote down these answers. Draw a diagram that could have been in the question for each one. a. a = 130c (E ’s at pt) b. b = 40c (E ’s on line) c. c = 80c (E sum of D) d. d = 35c (vert. opp. E ’s)
3 – 11 In each diagram, work out the marked angles. Give a brief reason from the table for each one.
E ’s on line vert. opp. E ’s E ’s at pt E sum of D
3.
4.
5.
6.
7.
8.
22
9.
10.
11.
Work out the size of the marked angles. 12.
13.
14.
15.
16.
17.
18.
19.
23
20.
21.
22.
23.
24. Here are two of the angles in some triangles. Write down the third angle each
time. a. {20c , 40c , } b. {45c , 95c , } c. {70c , 70c , } d. {83c , 90c , } e. {1c , 178c , } f. {60c , 60c , }
25. Which of the triangles in Question 24 would look most like a straight line? Draw a diagram to explain.
26. (Multichoice) Which of these sets of angles could not form a triangle? a. {40c , 60c , 80c} b. {105c , 15c , 60c} c. {73c , 48c , 59c} d. {35c , 64c , 71c}
For each diagram write down an equation with a geometrical reason. Then solve the equation to work out the value of x. 27.
28.
29.
30.
31.
32.
24
33.
34.
35.
36 – 37 Work out the sizes of each angle in these triangles. 36.
37.
Area of Simple Shapes Use the common area formulae to find the area of these figures.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
25
15.
16.
Composite Area
1.
2.
3.
4.
5.
6.
a. What fraction of the large
square has been shaded? b. What fraction of the large
square is unshaded?
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Find the area of each composite figure.
7.
8.
9.
10.
11.
12.
Application Problems using Area
1. A square 15 cm long and a rectangle 18 cm long have the same perimeter. Find the area of the rectangle.
2. Carlos is training for his 4.8 km run on a rectangular running track. If the running track measures 50 m wide and has an area of 7500 m2, how many rounds around the track must he run?
3. Lina has a rectangular piece of drawing paper 36 cm long and 21 cm wide. She pasted a rectangular picture in its center such that it is 3 cm away from its length and 8 cm away from its width. What is the area of the drawing?
4. A rectangular photo frame has a length of 24 cm and an area of 432 cm2. In its center is a rectangular piece of photograph 3 cm away from its length and 2 cm away from its width. Find the area of the photo frame which is not covered by the photograph.
5. A square and a rectangle have the same area. If the rectangle has a length of 32 cm and a perimeter of 80 cm, find the length of the square.
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6. A rectangular garden has an area of 486 m2. If it is 18 m wide, find the cost of fencing it at $16 per meter.
7. A rectangle is three times as long as it is wide. If it has a perimeter of 64 cm, find its area.
8. The perimeter of a square room is 36 m. What is the cost of tiling it at $8 per square meter?
9. A rectangular piece of land 78 m wide has a perimeter of 386 m. It has a pond with an area of 236 m2 in it. If the rest of it is covered with grass, what area of the land is covered with grass?
10. The diagram shows part of a pattern of tiles. Each tile is triangular.
a. Calculate the area covered by 1 tile. b. Calculate the area covered by 8 tiles.
11. A sheet of A4 paper measures 296 mm by 212 mm. It is printed with a 7 mm by 7 mm square grid for use in a Maths classroom. Estimate the number of squares on the sheet, to the nearest hundred.
12. The floor of a kitchen is rectangular, and measures exactly 3 m by 5 m. It is to be tiled with cork tiles measuring 250 mm by 250 mm. a. Will any of the tiles need to be cut into smaller pieces? Explain. b. Calculate the number of tiles that will be needed.
13. A reception lounge is rectangular in shape, and measures 20 m by 16 m. In the centre of the floor a rectangle measuring 6 m by 5 m has been tiled for dancing. The rest of the floor is carpeted. Calculate the area of carpet.
14. A chocolate bar is wrapped in a rectangular piece of foil measuring 10 cm by 15 cm. a. Calculate the area of the piece of foil. b. How many pieces could be cut out from a larger sheet of foil measuring 120 cm
by 75 cm?
15. DLE envelopes measure 22 cm by 11 cm. Which of the following is most likely to be the area of paper needed to make one of these envelopes? Explain. a. 242 cm2 b. 282 cm2 c. 484 cm2 d. 524 cm2
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16. A rectangular swimming pool has an area of 800 m2 and a width of 16 m. Calculate the length of the pool.
17. A farmer uses some fencing to construct a square pen to hold sheep for a competition. The area of the pen is 256 m2. What length of fencing will be needed to construct the pen?
Circle – Vocabulary Match the vocabulary word to the correct definition.
1. Radius a. the distance across a circle through its center
2. Circumference b. the distance around a circle
3. Center c. the point that is the same distance from all the points of a circle
4. Diameter d. the distance from the center to any point on a circle
5. Circle e. the set of all points in a plane that are the same distance from a given point
Find the circumference of the circle.
6.
7.
8.
Find the circumference of the circle described. Tell what value you used for r. Explain your choice.
9. d = 6 ft 10. d = 12 yd 11. r = 14 mm
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Area of a Circle Find the area of the circle.
1.
2.
3.
Find the area of the circle described.
4. r = 7 cm 5. r = 9 km 6. d = 10 yd
7. Find the area of a circle with a radius of 4.6 centimeters.
8. Find the area of a circle with a diameter of 25 millimeters.
Find the area of the figure.
9.
10.
11.
12. Find the shaded area. The figure shows two circles. The inner
circle has a radius of 8 cm and the circumference is 5 cm away from the circumference of the outer circle. (take r = 3.14)
13. Below are the semicircles of the above circles. Find the shaded area.
14. A dart-board is held together around the outside by a
metal band. One end of the band overlaps the other by 2 cm so it can be fixed on. The radius of the dartboard is 210 mm. Calculate the length of the metal band.
15. The radius of a bicycle wheel is 75 cm. How far has the bicycle travelled when the wheel has rotated once? True or false: ‘The diameter is the longest chord of a circle’.
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16. Find the shaded area of a.
b. c.
17. The diagram shows a circle with a radius of 2 cm inside a circle with a radius of 5 cm. Calculate the area of the shaded region.
18. Each of the squares here actually measures 4 cm by 4 cm. A
B
C
D
E
a. Which pattern has the smallest shaded area? b. Which pattern has the largest shaded area?
19. Compact disks (CDs) are usually packaged in square plastic cases. These cases measure 124 mm by 124 mm. A CD fits in the center of a case. There is a distance of 3 mm between the edge of the CD and the edge of the case. a. Sketch a diagram to show this information. b. Calculate the area of a CD (top only).
20. The diagram shows a triangle and a segment inside a circle. The radius of the circle is 8 cm. a. Calculate the area of the circle. b. Calculate the area of the triangle (A = 2
1 x b x h).
c. True or false: ‘Area of triangle + Area of segment = 41 x
Area of circle’? d. Calculate the area of the segment.
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21. Calculate the area of these sectors: a.
b.
22. A cell phone tower provides a clear signal for up to 6.25 km in any direction.
Accurately construct a circle using the scale 1 km = 1 cm and find the area that gets a clear signal.
23. Coach Dickinson is teaching his track and field athletes about the distance they can throw the discus. They know that if they throw the discus 18 metres, they have a good chance of being in the top 8 at the IASIS competition. Coach Dickinson tells them that the circular area that would be created by a throw of 18 metres is about 113 m2. Coach Lee disagrees and tells the athletes that the circular area created by a throw of 18 m is about 1017 m2. Who is correct? Explain both coaches’ calculations.
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Geometry – Blue Review Problems
1. A rectangular container, 25 cm long, 18 cm wide, and 10 cm high, contains 4 L 320 ml of water. Another rectangular container, 10 cm long, 6 cm wide and 4 cm high is completely filled with water. If water from the second container is poured into the first container, how much water will overflow?
2. In the figure shown, the circle and the square have the same perimeter. Find the difference in their areas. (Take r = 7
22)
3. The container shown is filled with water to a depth of 9 cm. How much water is it holding?
4. In the figure shown, ABC is a right-angled triangle and ADB and BDC are semi-circles of radii 18 cm. Find the area of the shaded parts. (Take r = 3.14)
A
B C
A
B C
5. In the diagram shown, how much more water can be poured into the container before it overflows?
6. The diagram shows the steel base for a vice. The measurements are in cm. Calculate the volume of steel used to make this.
33
Geometry – Black Problems
Plane Figures – Parallel and Perpendicular Lines
1. Classify each triangle, in as much detail as possible: a.
b.
c.
2. Decide whether the following pairs of triangles are congruent. State the test used, and if possible, write a congruence relationship between the triangles. a.
b.
3. Classify each quadrilateral given below: a.
b.
c.
4. Triangle ABC is isosceles with AC = BC. BC and AC are produced to E and D respectively so that CE = CD. a. Prove that DAEB and DBDA are congruent. b. What can you deduce about AE and BD?
5. Classify each figure giving brief reasons for your answer:
a.
b.
c.
34
6. Find the value of the pronumeral(s) for each figure, giving brief reasons for your answer: a.
b.
c.
d.
e.
f.
7. From each group of 3 triangles below, choose two triangles which are congruent. State why they are congruent and why the third triangle is not. a.
b.
8. In the rhombus PQRS given, prove that the diagonal QS bisects the angles at the vertices S and Q.
35
Facts and Terms from Geometry Complete these sentences. Choose the words that go in the gaps on the left from the list on the right.
1. A ____ is a line joining the centre of a circle to the circumference
180c 90c
2. Several points all on the same line are said to be ____ arc
3. Opposite angles in a cyclic quadrilateral add to ____ chord
4. A ____ is a line joining two points on a circle collinear
5. A quadrilateral which has all four vertices on a circle is said to be ____
concurrent
6. In a cyclic quadrilateral an interior angle is equal to ____
concyclic
7. Angles on the same arc of a circle are ____
cyclic
8. A ____ is part of a circle bounded by two radii and an arc
diameter equal
9. Points all on the circumference of the same circle are said to be ____
is twice
10. The angle in a semi-circle is ____
radius
11. A ____ is perpendicular to the radius at the point of contact.
sector segment
12. The angle at the centre of a circle ____the angle at the circumference
tangent
13. A ____ is a chord which passes through the centre of a circle
the ext opp E
14. An ____ is part of the circumference of a circle
15. Lines all of which pass through the same point are said to be ____
16. A ____ is part of a circle bounded by a chord and an arc.
36
Using Algebra in Geometry
Geometrical properties can be used to form equations and solve them to work out angles in a variety of situations. Examples In each case work out the angles marked x and y.
Answers 2x + 50c = x + 80c (ext E of D)
2x - x = 80 – 50 x = 30c
4y + (y + 70c) = 180 (opp E ’s cyc quad)
5y = 180 – 70 5y = 110 y = 22c
1. These diagrams show a pair of marked angles. Choose the relationship between the
angles from this list {alternate, corresponding, co-interior}. a.
b.
c.
2. Complete these sentences, using one of the terms:
alternate co-interior corresponding vertically opposite a. p and r are ____ b. p and s are ____ c. q and s are ____ d. p and q are ____
37
3. A transversal crosses two lines as shown in the diagram. For each pair of angles (a)-(f), say whether they are corresponding, alternate or co-interior. a. {b f} b. {b c} c. {e g} d. {g f} e. {c g} f. {f h}
4. Here are three different definitions about angles formed by a transversal that crosses two lines. Match up each one from the list {alternate, corresponding, co-interior}. a. This pair of angles are on opposite sides of the transversal, and between the
two lines. b. This pair of angles are on the same side of the transversal, and between the
two lines. c. This pair of angles are on the same side of the transversal, and both above or
both below the two lines.
5. Name the angle which is: a. alternate to q b. co-interior with v c. vertically opposite to w d. corresponding to u
Write the measures of the angles indicated in Problems 6 – 11. Do not use a protractor.
6.
m E r = ____ m E s = ____ m E t = ____
7. E JKL is a straight angle. m E NKO = ____
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8. m E a = ____ m E b = ____ m E c = ____
9. Angles a and t have the same measure.
m E a = ____ m E c = ____ m E t = ____
10. Angles x and y have the same measure. m E x = ____ m E y = ____ m E z = ____
11. m E p = ____
12. Give the value of the pronumeral(s) in each figure, with a brief reason for your answer: a.
b.
c.
d.
e.
f.
39
13. Find the value of the pronumeral(s) in each figure, giving a brief reason for your answer: a.
b.
c.
d.
e.
f.
Angle Properties
1.
2.
3.
4 – 14 Work out the sizes of the unknown marked angles in each diagram.
4.
5.
6.
7.
8.
9.
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10.
11.
12.
13.
14.
15. Copy and complete these sentences. Choose the missing words from this list:
{alternate, corresponding, co-interior}. Choose the missing letters from this list: {a, b, d, e, f, g, h}.
a. A pair of ____ angles on parallel lines map onto each other by translation. An
example is c and ____ b. A pair of ____ angles on parallel lines map onto each other by rotation about
O. An example is c and ____ c. A pair of ____ angles on parallel lines are supplementary. An example is c and
____
16. Are the pair of lines marked p and q parallel?
17. Which pair of lines is parallel?
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18. Write down the names of the parallel lines here.
19.
a. How many different pairs of alternate angles are there in this diagram? b. How many different pairs of corresponding angles are there in this diagram? c. How many different pairs of co-interior angles are there in this diagram? Plane Figures – Parallel and Perpendicular Lines – Angle Properties
1. Classify each figure giving brief reasons for your answer:
a.
b.
c.
2. Find the value of the pronumeral(s) for each figure, giving brief reasons for your answer: a.
b.
c.
d.
e.
f.
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3. From each group of 3 triangles below, choose two triangles which are congruent. State why they are congruent and why the third triangle is not. a.
b.
4. In the rhombus PQRS given, prove that the diagonal QS bisects the anglet the vertices S and Q.
In each of these questions form an equation, state the geometric property you have used, and solve it to work out x.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
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16. Write down an equation linking angles a, b, c and d.
17. a. Form an equation and solve it to work out the four angles of this quadrilateral (not drawn to scale).
b. Explain why the quadrilateral is not cyclic. c. What kind of quadrilateral is it?
18. Calculate angle a given a = b + 20 b = c + 20 c = d + 20
19. Calculate angle x. The large triangle ABC is isosceles, with AC = BC
Answer each of the following questions. Show your work and write your statements clearly. 20. In the figure shown, 0 is the center of the semi-circle. Find E s.
21. In the figure shown, ABCD is a parallelogram. Find E t.
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22. In the figure shown, EFG and EHI are isosceles triangles. Find E v and E w.
23. In the figure, not drawn to scale, XYZ is an isosceles triangle. Find E c.
24. In the figure, not drawn to scale, ABCD is a rectangle. Find E d.
25. The figure shows a regular octagon. Find E m and E n.
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Area and Circumference of a Circle Find the area of each shape showing all working.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. Compact disks (CDs) are usually packaged in square plastic cases. These cases measure 124 mm by 124 mm. A CD fits in the center of a case. There is a distance of 3 mm between the edge of the CD and the edge of the case. a. Sketch a diagram to show this information. b. Calculate the area of a CD (top only).
11. The diagram shows a triangle and a segment inside a circle. The radius of the circle is 8 cm. a. Calculate the area of the circle. b. Calculate the area of the triangle (A = 2
1 x b x h).
c. True or false: ‘Area of triangle + Area of segment = 41 x
Area of circle’? d. Calculate the area of the segment.
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12. A cell phone tower provides a clear signal for up to 6.25 km in any direction. Accurately construct a circle using the scale 1 km = 1 cm and find the area that gets a clear signal.
13. Coach Dickinson is teaching his track and field athletes about the distance they can throw the discus. They know that if they throw the discus 18 metres, they have a good chance of being in the top 8 at the IASIS competition. Coach Dickinson tells them that the circular area that would be created by a throw of 18 metres is about 113 m2. Coach Lee disagrees and tells the athletes that the circular area created by a throw of 18 m is about 1017 m2. Who is correct? Explain both coaches’ calculations.
14. The radius of circle O is 8 cm. How many centimeters are in the length of the diagonal of the rectangle?
15. John ties the leash of his dog to the corner of the doghouse. The dimensions of the dog house are 3’ x 4’, and the leash is 6’ long. Over how many square feet can John’s dog wander? Express your answer as a decimal to the nearest hundredth.
16. A regular hexagon is inscribed in a circle. If the perimeter of the hexagon is 42 inches, how many inches are in the circumference of the circle? Express your answer in terms of r .
17. In a race, athletes run three laps around an oval race track formed by a rectangle and two semicircles as shown. The length of a radius of each semicircle is 12 meters, and the length of the rectangle is twice the length of the diameter of the semicircle. What is the number of meters in the length of the race? Express your answer in terms of r .
18. A square dartboard has an inscribed circle. A thrown dart hits the square target. What is the probability that the dart lands outside of the inscribed circle? Express your answer as a common fraction in terms of r .
19. The three circles are concentric with center C. On segment CF, CD = 3 cm, DE = 2 cm, and EF = 1 cm. The picture shows three enclosed, non-overlapping regions. What is the number of square centimeters in the area of the largest region? Express your answer in terms of r.
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The Pythagorean Theorem
Example 1: Find c the length of the hypotenuse, in this triangle.
c 2 = a 2 + b 2 c 2 = 6 2 + 8 2 c 2 = 36 + 64 c 2 = 100 c = 100 Example 2: Find the value of x in the triangle at the right. Round to the nearest tenth. a 2 + b 2 = c 2 Use the Pythagorean Theorem. 6 2 + x 2 = 92 Replace a with 6, b with x, and c with 9. 36 + x 2 = 81 Simplify. x 2 = 45 Subtract 36 from each side. x = 45 Find the positive square root. Then, use one of the two methods below to approximate 45 . Method 1 Use a calculator. A calculator value for 45 is 6.708203932. x . 6.7 Round to the nearest tenth. Method 2 Use a table of square roots. Use the table on page 746. Find 45 in the N column. Then find the corresponding value in the N column. It is 6.708. x . 6.7 Round to the nearest tenth. The value of x is about 6.7 in.
1. Investigate:
On graph paper, create right triangles with legs a and b. Measure the length of the third side c with another piece of graph paper. See illustration.
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2. Complete the table below. a b c a 2 b 2 c 2 3 4 9 16 5 12 144 9 12 144
3. Based on your table, use > or < or = to complete the following statement.
a 2 + b 2 ? c 2
4. In any right angle triangle, the sum of the squares of the legs is equal to the square of the lengt5h of the hypotenuse. a 2 + b 2 = c 2
5 – 9 Use the Theorem of Pythagoras twice to calculate the lengths marked x. Give your answers accurate to 4 sf.
5.
6.
7.
8.
9.
10 – 14 Form an equation and solve it to work out the side lengths of these right-angled triangles. 10.
11.
12.
13.
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14. The distance from the mid-point of a chord which is 7 cm long to the centre of a
circle is 8 cm. Calculate the radius of the circle.
Can you form a right triangle with the three lengths given? Show your work. 15. 4 m, 6 m, 7 m 16. 5 cm, 12 cm, 13 cm 17. 7 in., 24 in., 25 in.
18. 1 ft, 3 ft, 12 ft 19. 4 mi, 5 mi, 6 mi 20. 1 m, 0.54 m, 0.56 m
21. 8 in., 10 in., 12 in. 22. 5 yd, 3 yd, 2 yd 23. 3p ft, 4p ft, 5p ft
Use the triangle at the right. Find the missing length to the nearest tenth of a unit. 24. a = 2 in., b = 4 in., c = 25. a = 1.4 m, b = 2.8 m, c =
26. a = 3 ft, c = 5 ft, b = 27. b = 2.7 km, c = 3.4 km, a =
Algebra. Find the value of n in each diagram. Give your answer as a square root. 28.
29.
30.
31.
Problem Solving Using the Rule of Pythagoras
• Draw a neat, clear diagram of the situation. • Mark down lengths and right angles on the diagram. • Use a symbol, such as x to represent the unknown length. • Write down the Pythagoras rule for the given situation. • Solve your equation • Write your answer in a sentence where necessary.
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Applications of Pythagoras
1. Application Problem. A rectangular field has length 110 m and width 50 m. Jackson runs along the length of the field then across the width and finally along the diagonal. What is the total distance he runs?
2. Application Problem. A sports instructor required her players to start at point A and either run all the way around the field twice or six times up and down the diagonal. Which is further and by how much?
3. A cyclist rides 8 km due west and then 10 km due north. How far is he from his starting point?
4. Two ships B and C leave airport A at the same time. B travels in a direction 067c at a constant speed of 36 km/h. C travels in a direction 157c at a constant speed of 28 km/h. Find the distance between them after 2 hours.
5. A baseball ‘diamond’ is a square whose sides are 27 m long. Find, to the nearest 10
1 metre, the distance from the
home plate to second base.
6. To protect against earthquake damage all new houses are required to have suitable diagonal bracing in the walls as specified in the Building Code.
7. The escalator at a shopping mall rises 4.3 metres from the ground level to the level above. By travelling on this escalator a shopper moves 13.9 m horizontally. Calculate the diagonal length of the escalator.
8. After a rugby test, John and Matiu have to go from one corner to the opposite corner of the rectanguler field. They can walk at 2 m/s. The field measures 56 m by 112 m. How much time would they save by walking diagonally across the field compared with walking along two sides to reach the opposite corner?
9. The lounge in Rupert’s house measures 8.1 m by 5.1 m. Every year at Christmas Rupert stretches a piece of string diagonally across the lounge at the same height to display his Christmas cards. The length of the piece of string is x metres. Represent this information on a diagram.
10. From a distance a wigwam tent looks like an isosceles triangle. The base of the wigwam is 6 metres, and the distance from the top to the ground by the left of the wigwam is 7 metres. The height of the pole that stands directly underneath the top of the wigwam is x metres. Draw a diagram that shows this information.
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Extensions of Pythagoras Pythagoras examples can involve more than one right-angled triangle. When working out this kind of problem do not round the answers to intermediate calculations until the very end, or you will lose accuracy. Example Calculate the length marked x.
Answer To work out x we need to use the length marked AC. AC 2 = 82 + 52 = 64 + 25 = 89
AC = 89 = 9.434 (4 sf) Now use Pythagoras again in DACD: x 2 = AC 2 + 6 2 = 89 + 36 = 125 => x = 125 = 11.18 (4 sf) When you square a square root of a number, you get back the original number – for example 89` j
2 = 89. But if 9.434 2 is used,
accuracy is lost: 9.434 2 = 89.000 356
The Theorem of Pythagoras can be found in many areas of mathematics – for example in quadratic equations and circle geometry. Example Calculate the sides of this triangle.
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Answer PQ 2 + QR 2 = PR 2 {Pythagoras} x 2 + (x + 7) 2 = (x + 8) 2 => x 2 + x 2 + 14x + 49 = x 2 + 16x + 64 => 2x 2 + 14x + 49 = x 2 + 16x + 64 => x 2 - 2x - 15 = 0 => (x - 5)( x + 3) = 0 => x = 5 or -3 (reject -3 because lengths are not negative) The sides are 5, 12 and 13 Example The mid-point of a chord of length 10 cm is 3 cm from the centre of a circle. Calculate the length of the radius. A radius always bisects a chord at right-angles.
Answer First draw a diagram of the circle, and the right-angled triangle found in it.
r2 = 5 2 + 3 2 = 25 + 9 = 34 r = 34 = 5.831 (4 sf)
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1. Calculate the length of the chord AB.
2. A chord of length 15.8 cm is drawn inside a circle which has a radius of 19.2 cm. Calculate the distance from the centre to the middle of the chord.
3. The diagram shows that tangents drawn from a point T to a circle touch it at A and B. The centre of the circle is 0. Calculate the distance CT from the point T to the circle.
4. A tangent is drawn from a point P to a circle, touching the circle at T. The radius OT of the circle is 12 cm, and the distance PT is 37 cm. Show this information on a sketch diagram. Calculate the distance from P to the centre of the circle.
5. The diagram shows Oliver (a Roundhead) wearing a dunce’s cap. The distance between Oliver’s ears is 220 mm, and it is 430 mm from the point of the cap to where the cap touches his head, just above his ears. Calculate, to the nearest mm, the distance from the point of the cap to the top of Oliver’s head.