if the transversal cuts two parallel lines then...

Unit 6 – Measurement and GeometryDate Topics Assignment

Completed

6.1 Pythagorean Theorem

6.2 Area and Perimeter of Composite Shapes

6.3 Part 1 – Volume of Prisms

6.3 Part 2- Volume of Pyramids & Cones

6.4 Part 1 - Surface Area

6.4 Part 2 - Surface Area

6.5 Angle Relationship in Triangles and Quadrilaterals

6.6 Angle Relationships in Parallel Lines

6.7 Interior Angle Relationships in Polygons

6.8 Exterior Angles Relationships in Poylgons

Measurement and Geometry Assignment.

6.1 The Pythagorean Theorem

Investigate: 1. What type of triangle is in the centre of the picture?

2. Complete the table below

Side a = Area of A =

Side b = Area of B =

Side c = Area of C =

3. How is the area of A and the area B related to the area of C?

4. Name the famous mathematician who first figured this out.

RULE:Pythagorean Theorem states …

Example 1:

a) Label the hypotenuse a) Label the hypotenuse

b) Calculate the missing side length b) Calculate the missing side length

10 cm

11 cm

9 mm 7 mm

Example 2: A ladder, which is 8.5 m long, leans against a wall. The foot of the ladder is 2.3 m from the base of the wall. How far up the wall does the ladder reach?

Example 3: The ranger has to search the area of the outlined in the triangle.Determine the perimeter and area that the ranger must cover.

BeachCabin

3 km 2.4 km

Ranger Station

Example 4: Determine the height of the cone.

19 cm

8 cm

40 cm

30 cm

3.2m

9.8m

7Km 6Km

10Km

B

A C

D

Assignment 6.1 The Pythagorean Theorem

1. Solve for the missing side in the following triangles:

a) b)

2. A wall is supported by a brace 10 feet long, as shown in the diagram below. If one end of the brace is placed 6 feet from the base of the wall, how many feet up the wall does the brace reach?

3. When Arnold swims laps in his rectangular swimming pool, he swims along the diagonal so he doesn’t have to turn around so often. Find the distance Arnold travels by swimming once along the diagonal.

4. A baseball diamond is a square with sides 90 ft. What is the distance to the nearest tenth of a foot between 1st base and 3rd base?

5. In a computer catalog, a computer monitor is listed as being 19 inches. This distance is the diagonal distance across the screen. If the screen measures 10 inches in height, what is the actual width?

6. Consider a right triangle that has a height of 5 and a base of 12. a. Find the length of the hypotenuseb. Find the perimeter of the trianglec. Find the area of the triangle

7. Towns A, B, C, and d are situated as shown on the diagram.a. How far is it from town B to town D?b. How far is it from town B to town C?

8. Determine the value of x and y.

9. The length of each of the three sides of a triangle is given. Determine whether the triangle is a right triangle. Justify your answers using calculations.

a) 5 cm, 8 cm and 9 cm b) 18 cm, 24 cm and 30 cm

10. Determine the height of the trapezoid

11. Determine the value of x and y

1 a. 50cm b. √297 2. 8ft 3. 10.31m 4. 127.28ft 5.√261 6a. 13cm b.30cm c. 30cm2 7a. √51km b. √87km 8. y=20 x=25 9. No – proof required

9b. Yes – proof required 10. 4cm2 11. x=√119 y=√443

6.2 Perimeter and Area of Composite Shapes

Warm Up:

Oscar has the choice between four pizzas below. Determine which pizza has the greatest area. Show your work.

Ex 1) A field consists of a rectangle and a half-circle, as shown below. What is the perimeter of the field? Show your work.

Ex 2) The figure below is made up of a triangle and a rectangle. Determine the area of the figure. Show your work.

Ex 3) What is the area of the shaded region on the figure below? Show your work.

Assignment 6.2 Perimeter and Area of Composite Shapes

1. Calculate the perimeter and area of the figure below (to 2 decimal places).a) b) c)

2. Find the area of the shaded and un-shaded regions. Show your work.a) b)

3. Use the figure on the right to answer the questions.a) What is the area of the square?

b) What is the area of the triangle on the left?

c) What is the area of the composite figure?

4. Determine the perimeter and area of the following figure. Show your work.

a) b)

5. The figure below is made two semicircles. What is the area of the shaded region? Show your work.

6. The area of the larger rectangle below is of 1120 m2. What is the area of the shaded rectangle? Show your work.

7. A figure is a triangle composed of and a square, as shown below. The area of the square is 144 m2. What is the height of the triangle? Show your work.

8. Determine the perimeter of the figure below. Show your work.

Solutions: 1.a) P = 86.2m A = 325cm2 b) P= 48.54cm A= 110.13cm2 c) P= 20.8 A = 21 units2

2a) Aun-shaded = 28.27m2 Ashaded = 7.73m2 b) A un-shaded= 72 units2 A shaded = 72 units2 3. a) 25 units2 b) 15 units2 c) 55 units2 4. a) P = 50 cm A = 160cm2 b) P = 38m A = 46m2

5. 131.95cm2 6. 192m2 7. 6.7m 8. 38.8cm

length

height width

12 cm

7cm5cm

6.3 Part 1 - Volume of Prisms & Spheres

Example:

Calculate the volume of the prism on the right.

The base of the prism is the side that is sitting on the ground. This shape is a rectangle with length 5 cm and width 7 cm.

Because we are multiplying 3 dimensions, volume is expressed in “cubic” units. For example: cm3, or m3. Make sure that your answer above has the correct units.

Example 2) Calculate the area of the base, and then calculate the volume.

a) b)

A prism is a 3-dimensional object because it has 3 dimensions:

length, width and height.

Note: Note:

To find the volume of a prism, find the base area and multiply by the height

Volume of a Sphere

Spheres are very unique shapes and as such, it is very difficult to figure out exactly how to find the volume of a sphere. Fortunately for us, Archimedes, the Greek mathematician already figured it out a long time ago.

The volume of a sphere is given by the following formula:

V =

Ex. Give the following sphere, determine its volume.

Ex 2. Calculate the volume of the following figures

a) b)

5.1 cm

5.1 cm

5.1 cm

Assignment 6.3 Part 1- Volume of Prisms & Spheres*answer on SEPARATE piece of paper. Include formulas & full solution

1. Calculate the volume of each prism (round to nearest tenth)a) b)

c)

2. Calculate volume of each sphere (round to nearest tenth)a) b)

3. A solid sphere is placed inside of the box shown. How much empty space will be left over in the box ?

4.5.6.7.8.9.

2.5 cm a) What is the volume of the box?b) What is the volume of the sphere?c) How much empty space will there be in the box?

4. The Old MacDonald playset has a silo as shown below, made of a cylinder and a hemisphere. What is the volume of this solid?

5. For F.R.O.S.T.Y, the grade nine class is selling Toblerone bars as shown below that has a filled chocolate volume of 80 cm3. We need to know the length of wrapping paper that would fit the bars. What is its height, h?

Answers: 1.a) 39m3 b) 769.7cm3 c) 1902.8 cm3

2.a) e) 3053.6 cm3 b) 268.1 cm3

3.a) Vbox = 5.13 b) Vsphere = 65.4cm3 c) 67.3cm3 4. 1204.3cm3 5. 8cm

6.3 Part 2 - Volume of Pyramids & Cones______________________________________________________________

Volume of Pyramids and Cones

If we have a rectangular prism and a similar pyramid, with a base of the same area and the having the same height… The volume of a pyramid is 1/3 the volume of the related prism.

FORMULAS: Volume of rectangular prism= _________________

Volume of a rectangular pyramid = ____________

Key terms:

Height (always at 90 degree angle)(altitude)

Slant height (hypotenuse)

How would Pythagorean Formula help?

Volume of Cones and CylindersThe same relationship is true of cones and cylinders. If we have a similar pair:

Examples:a) b) c)

A cone has volume 263.9 cm3 and base radius 6 cm. What is the height ofthe cone?

5 m

3 m 24 m

13 m

24 m 25 m

23 m

25 m

15 in.

18 in.

Assignment 6.3 Part 2- Volume of Pyramids & Cones*answer on SEPARATE piece of paper. Include formulas & full solution

1. Determine the volume of each solid. Round to the nearest tenth.a) b) c)

d) e)

2. Determine the height of a cone if the volume of the cone is 56cm3 and the diameter is 5cm. Round your answer to the nearest tenth.

3. The volume of the square pyramid below is 720 cm3. What is the height of the pyramid?

4. If a solid cylinder is placed inside a cone ( shown to the right), how much empty space is inside cone?

Solutions:1. a) 4791.7m3 b) 960 m3 c) 37.7 m3 d) 1017.9 in3 e) 882 ft3

2. 8.6cm3 3. 15cm 4. 2799.2 cm3

6.4 Part 1 - Surface Area of Prisms

Remember that when a 2D net is folded together it turns into a 3D shape. Surface Area requires finding the area of ALL surfaces of the shape.

Example 1) Find the surface area of the rectangular prism

(a)

Example 2) Find the area of the following cylinder.

Example 3) The radius of the following sphere is 28 cm. What is its surface area?

Example 4) A ball has a surface area of approximately 9900 cm2. What is its radius?

Assignment 6.4 Part 1 - Surface Area of Prisms

1. Aaron is building a storage chest in the shape of a rectangular prism. The chest will be 90 cm long, 70 cm deep, and 60 cm high. What will be the outer surface area of the box?

2. Alex is going to construct a fish tank that is 1.2 m long, 0.6 m wide, and 0.4m high. How much glass will he need to make it? (Note: There will be no glass on the top.)

3. Find the surface area of a cylindrical tank that has a radius of 1.5 m and a height of 5 m.

4. Find the surface area of a sphere with a radius of 1.3 m.

5. Find the surface area of a sphere with a diameter of 24.8 mm.

6. Find the surface area of a hemisphere with a radius of 18.5 cm.

7. A tennis ball has a diameter of 6.7 cm. What is its surface area?Pauline removes the label, which covers the cylindrical soup container. The box has a volume of 785 cm3 and a height of 10 cm. Find the area of the label removed by Pauline.

8. The solid compound is below a cylinder and a hemisphere. What is the surface area of this solid?

Answers: Check answers!!!

1. 31800 cm2 2. 2.16 m 2 3. 35.3m 2 4. 21.2m 2 5. 1932.2mm 2 6. 2150.4cm 2 7. 141.03cm 2

16. 314.2cm 2 17. 691.1cm2

6.4 Part 2 - Surface Area of Pyramids & Cones

Surface area of PyramidsTo determine the surface area of a pyramid, we need to find out the SA of the base and then the 4 sides.

Example

Find the surface area of the square-based pyramid.

(a)

Slant Height = 16m

(b)

Surface Area of ConesA cone is like a __________________, except it has a circular base. The surface area of the lateral area of the cone (the area not including the base) is calculated with the formula:

Example (a) Find the surface area of a cone that has a radius of 12 feet and slant height of 15 feet.

(b) Katherine wants to create a cone in the following shape. She would like to paint the cone red, but does not want to paint the top. How much area does Katherine need to paint?

Assignment 6.4 Part 2 -Surface Area of Pyramids & Cones

1. Find the surface area of the square-based pyramid to the right.

2. Find the total surface area of a square pyramid with a base of 12 cm by 12 cm and a height of 8 cm.

3. If the surface area of the sides of a square-based pyramid is 680 cm2

and the side lengths of the square are 16 cm, what is the height of the pyramid?

4. Find the surface area of a cone that has a slant height of 82 cm and a radius of 28 cm.

5. Find the surface area of a cone with a diameter of 13.6 cm and a slant height of 9.8 cm.

6. Find the total surface area of a cone with a radius of 16 inches and a height of 20 inches.

7. Find the surface area of a cone with a radius of 45.7 mm and a height of 39.7 mm.

8. Find the lateral surface area of a cone whose height is 32 cm and whose diameter is 28 cm.

Answers:1. 336cm2 2. 384 cm2 3. 13.25cm 4. 9676.1 cm2 5. 354.6 cm2 6. 2091 in2 7. 15247.2mm 2

8. 2150.7cm2

6.5 Angle Relationship in Triangles and Quadrilaterals

Warm Up:

Opposite Angle Theorem (OAT)- opposite angles are ____________

Complementary Angles (CA)- two angles add up to ______________

Supplementary Angles (SA)- two angles add up to _________

Angle Sum Triangle Theorem (ASTT)- three angles in a triangle add up to _________

Isosceles Triangle Theorem (ITT)- angles opposite the equal sides are _________

Equilateral Triangle Theorem (ETT)- all three sides equal and all three angles ________

Exterior Angle Theorem (EAT)- exterior angle of a triangle is equal to the sum of the interior and non-adjacent angles

Angle Sum Quadrilateral Theorem (ASQT)- four angles in a quadrilateral add up to _________

Parallel Line Theorem (PLT)- when a transversal crosses two parallel lines then :

corresponding angles are ______________________________

(PLT – F pattern)

alternate angles are _________________________

(PLT – Z pattern)

interior angles are ______________________________

(PLT – C pattern)

67° 44°z130° y32°

x

125° x

y

57°

c

b

a

35°70°

x

62

k

36 36

b

95

2x

3xx

Grade 8 Review:

1) Determine the measure of each indicated angle. State your reasoning.a) b) c)

d) e) f)

g) h)

2) Determine the value of x, and then the measure of each angle

95

2y

30

y

d

105

95

Exterior Angles of Triangles and Quadrilaterals

The exterior angles of a triangle add to ____________The exterior angles of a quadrilateral add _______________

Proof:

Practice ProblemsDetermine the size of each indicated angle.

a) b)

c x

82°

33°

z r

t

37°72° 47°

x

39°118°

y

x

121 35

x

127

100

85 b

a

101

112

81

x

// \\

55 x

y 97

8276

d

cb

ea

51 117

x y

z

X+35 x-35

xA

B

3x x

2x

B

C

A

x

4x 3x

2xA

B C

D

3010 x205 x

106 x

Assignment 6.5: Angle Relationship in Triangles and Quadrilaterals

1) Determine the measure of each indicated angle. Justify your answers using geometric properties.

a) b) c)

d) e) f)

g) h) i)

2) Find the value of x, then find the measure of all of the angles. Justify your answers using geometric properties.

a) b) c)

d) e)f)

C

A A ABB

B

C C

D

CD

E

x y60°

z

z

40°

y28° x

105°

x

y72°

28°

yz95°

35°

x

154° y

37°

z x

51°

3. Determine the value of the variables. Justify your answers using geometric properties.

a) b)

c) d)

e) f)

5. Pravin designs a lightning bolt using two quadrilaterals and one triangle as shown below. Determine the value of x and y. Justify your answers using geometric properties.

6. Determine the value of x in the diagram below. Justify your answers using geometric properties.

6.5: Answers1) a) x = 61º b) x = 62º , y = 79º c) z = 65º , r = 65º , t = 78º d) x = 24º e) x = 66º f) a = 122º , b = 53º g) x = 66º , y = 114º , z = 129º h) a = 83º , b = 98º , c = 104º , d = 75º , e = 105º i) x = 55º , y = 70º 2) a) x = 30º . <ABC = 90º , <CAB = 30º , <ACB = 60º b) x = 60º , <ABC = 95º , <CAB = 60º , <ACB = 25º c) x = 36º , <ABC = 144º , <BCD = 108º , <CDA = 72º , <DAB = 36º d) x = 56º, <A = 116, <B = 44, <C = 65, <D = 135 e) x = 20, <A = 170, <B = 110, <C = 80 f) x = 94, <AEB = 112, y = 34, <BEC = 68 3) a) x = 62, y = 35, z = 145 b) x = 30, y = 150, z = 30 c) x = 50, y = 40, z = 85 d)x = 152, y = 18 e) x = 39, y = 26, z = 53 f) x = 130 4) x = 93, y = 111

6.6 Angle Relationship in Parallel Lines

Term Definition Diagram

Parallel Lines

Transversal

Corresponding Angles

Alternate Angles

Interior Angles

If the transversal cuts Two Parallel Lines then the PARALLAL LINE THEOREM (PLT) states

Alternate angles ___________________.

We call this the PLT Z-Pattern.

Corresponding angles _________________.

We call this the PLT F-Pattern.

Interior angles ___________________.

We call this the PLT C-Pattern.

A

B

CD

E

F

75o

w x

y

z

64o

c

a

49o

98o

mn p

Practice Problems1. Determine the angle measure indicated by each small letter. Justify your answers using geometric properties.

a)

b)

c)

Statement Reason

Statement Reason

Statement Reason

G

H

Sometimes we need to determine the measures of other angles in order find the one we are actually looking for.

2) Determine the measure of the unknown angles below. State all your geometric reasoning.

a) b)

Sometimes we need to create an equation in order to determine the missing angle.

3) Determine the measure of the unknown angles below. State all your geometric reasoning.

a) b)

c) d)

q

r

112

s

75

y x

51 72

y

x 35

75

z y

x

118 xy

87 z

30

44

c

b

a

Assignment 6.6 Angle Relationship in Parallel Lines

Determine the angle measure indicated by each small letter algebraically. Do not use a protractor. Justify your answers using geometric properties.

1) 2)

3) 4)

5) 6)

7) 8)

9) 10)

c117° b

a

de

64°de

f

g

113°143°

h i j

k

33°

75°

l

11) Write an equation and solve for the unknown. State the theorem used to make the equation.a) b)

c) d)

e) f)

g) h)

6.6 ANSWERS:1. y = 75o, x = 105o 2. r = 112o, s = 112o, q = 68o 3. x = 35o, y = 70o, z = 75o 4. x = 51o, y = 57o, 5. a = 44o, b = 30o, c = 106o 6. x = 62o, y = 87o, z = 93o 7. a= 117, b = 63, c =63, d = 63, e = 117 8. e = 116, d = 64, f = 116, g = 116 9. i = 76, h = 37, j = 67 10. l = 72, k = 33 11. a = 30, b = 110, c = 40, d = 30, e = 60, f = 134, g = 28, h = 32

6.7 Interior Angles Relationship in Polygons

Investigate how to determine the sum of the interior angles of a polygon by completing the following table.

Polygon Number of Sides Sketch of Polygon Number of

TrianglesSum of Interior

Angles

Triangle 3 1 180°(180° x 1)

Quadrilateral 4 2 360°(180° x 2)

Pentagon 5 3 540°(180° x 3)

Hexagon 6

Heptagon 7

Octagon 8

n-gon n

Practice Problems: Interior Angles of a Polygon

1. Find the sum of the interior angles of each polygon.

a) b) c)

2. Find the sum of the interior angles of each regular polygon. a) b) c) d)

3. Define a regular polygon.

4. Can you determine the measure of each interior angle in question 1? Why or why not?

5. Can you determine the measure of each interior angle in question 2? Why or why not?

6. Determine the measure of each interior angle in question 2?

7. Find the sum of the interior angles of a polygon with each number of sides.a) 18 sides b) 24 sides

8. Find the measure of each interior angle of a regular polygon with each number of sides.a) 20 sides b) 16 sides

9. Find the number of sides each polygon has given the sum of its interior angles.a) 2340 b) 4140

10. Determine the measure of the missing angles. Justify your answer using geometric properties.

a) b)

Assignment 6.7 Interior Angles Relationship in Polygons

1. Find the sum of the interior angles in a polygon witha) 8 sides b) 12 sides

2. How many sides does a polygon have if the sum of the interior angles is a) 1440° b) 2520°

3. Determine the value of x. Justify your answers using geometric properties.

a) b) c) d)

4. Determine the value of x, and then determine the measure of each interior angle. Justify your answers using geometric properties.

a) b)

5. In baseball the home plate is shaped like the one shown. It has 3 right angles and 2 other congruent angles (A and B). Find <A and <B. Justify your answers using geometric properties.

BA

85°

130°

120°

150°

130°

g q

edcp

b

a

f

6. Determine the values of each of the unknowns in the diagram below. Justify your answers using geometric properties.a) b)

7. Determine the values of each of the unknowns in the diagram below. Justify your answers using geometric properties.

6.7 Answers: 1. a) 1080° b) 1800° 2. a) 10 b) 16 3. a) 96 b) 76 c) 102 d) 70 4. a) x = 110 110, 145, 120, 95, 130,120 b) x = 72 144, 108,108, 90, 90 5. Both A and B are 135 6. a) x = 45 b) x = 36 7. a = 95, b = 85, c = 80, d = 150, e = 130, f = 50, g = 130, p = 115 q = 140

6.8 Exterior Angles Relationship in Polygons

Let’s investigate how to determine the sum of the exterior angles of a polygon

Part A) Exterior Angles of a Quadrilateral Draw a large quadrilateral. Label the vertices. Estimate the measure each of the four interior angles.

Extend one side at each vertex of your quadrilateral to create an exterior angle. Name and measure the four exterior angles. Find the sum of these exterior angles. Compare this sum to those found by your classmates.

Make a hypothesis about the sum of the exterior angles of any quadrilateral.

Key Concepts For a polygon with n sides, the sum of the interior angles, in degrees, is

_______________. The sum of the exterior angles of a polygon is _____________________.

Practice Problems1. Consider a regular 20-sided polygon, what is the measure of …..

a) The sum of the interior angles b) The sum of the exterior angles

c) Each interoir angle d) Each exterior angle

2. Six exterior angles of a 7-sided polygon each measure 50°. Find the measure of the missing angle.

3. Each exterior angle of a regular polygon is 12˚. Determine the number of sides.

4. A stop sign in the shape of a regular octagon is resting on a brick wall, as shown in the accompanying diagram. What is the measure of angle x?

6. Determine the value of each exterior angle in the diagram below.

Assignment 6.8 Exterior Angles Relationship in Polygons

1. Find the measure of each exterior angle in a regular polygon witha) 7 sides c) 9 sides

2. Find the measure of each interior angle in a regular polygon witha) 5 sides b) 6 sides

3. Determine the measure of x. Justify your answers using geometric properties.

a) b)

4. Melissa is walking around the outside of a building that is in the shape of a regular polygon. She determines that the measure of one exterior angle of the building is 60°. How many sides does the building have?

5. Determine the value of x, then determine the value of each exterior angle. Justify your answers using geometric properties.

6. An exterior angle of a regular polygon measures 72o. What is the measure of its corresponding interior angle?

7. What is the measure of one interior angle of a regular 40-gon?

8. What is the measure of one exterior angle of a regular 40-gon?

9. If an exterior angle of a regular polygon measures 45 o, how many sides does the polygon have?10. The five exterior angles of a pentagon measure 6x+2, x-3, 3x-5, and 2x+7, and x+11. Find the measures of the five angles.

6. 8 Answers:1.a) 51.43° b) 40° 2.a) 108° b) 120° 3. a) 36° b) 60 4. 6 sides 5. x = 40 ; angles: 90, 65, 70, 80, 556. 108 7. 171 8. 9 9. 8 sides 10. X = 27 angles: 164, 24, 76, 61, 38

xx

x

xx

x

x

x

x

x

x