8th grade geometry

Example

A ladder leans against a wall. Its foot is 8 m from the wall and it reaches 3 m up the wall. How long is the ladder? Round to 1 decimal place.

1. 8² + 3² = 73
2. √73 = 8.544…
3. ≈ 8.5 m

Answer: 8.5

Example

Find the midpoint of (3, −5) and (3, −3). Type it like (2, −1).

1. x = (3 + 3) ÷ 2 = 3
2. y = (−5 + (−3)) ÷ 2 = −4
3. (3, −4)

Answer: (3, −4)

Example

A line crosses two parallel lines. One angle is 32°. What is the corresponding angle to it, in degrees?

1. Corresponding angles are equal: 32°.

Answer: 32

Example

What is the sum of the interior angles of a polygon with 12 sides, in degrees?

1. (12 − 2) × 180° = 1800°

Answer: 1800

Example

A cylinder has radius 8 yd and height 8 yd. Find its volume in yd³, rounded to 1 decimal place.

1. π × 8² × 8 = 512π
2. ≈ 1608.5 yd³

Answer: 1608.5

Example

Diego says the hypotenuse of a right triangle with legs 6 and 8 is 14, because you add the legs. What is it really?

1. √(6² + 8²) = √100 = 10

Answer: 10

Example

A right triangle has hypotenuse 17 cm and one leg 4 cm. Find the other leg, to 1 decimal place.

1. 289 − 16 = 273
2. √273 ≈ 16.5 cm

Answer: 16.5

Example

A triangle has sides 6, 8 and 10. Is it a right triangle?

1. 36 + 64 = 100
2. 10² = 100
3. Equal: right triangle.

Answer: Yes

Example

A rectangular park is 110 m long and 67 m wide. Instead of walking along two sides, you cut across the diagonal. How many metres shorter is the shortcut? Round to the nearest metre.

1. √(110² + 67²) ≈ 128.8 m
2. 177 − 128.8 ≈ 48 m

Answer: 48

Example

Triangle PQR has P(0, 0), Q(4, 0), R(0, 4). It is translated by (−2, −4). Where does R go?

1. (−2, 0)

Answer: (−2, 0)

Example

Dilate (−3, −5) by a scale factor of 4 with centre the origin. Type the image like (2, −3).

1. (−12, −20)

Answer: (−12, −20)

Example

Which transformation is described by (x, y) → (x, −y)?

1. Reflection in the x-axis

Answer: Reflection in the x-axis

Example

Two triangles are similar. The small one has sides 7 cm and 8 cm. The matching side to 7 cm on the large triangle is 14 cm. How long is the side matching 8 cm?

1. k = 2
2. 8 × 2 = 16 cm

Answer: 16

Example

A transversal crosses two parallel lines. Two angles are in the same position at each crossing. What are they called?

1. They are corresponding angles.

Answer: corresponding

Example

Alternate angles on parallel lines are (3x + 21)° and (x + 43)°. Find x.

1. 3x + 21 = 1x + 43
2. 2x = 22
3. x = 11

Answer: 11

Example

An exterior angle of a triangle is 106°. One of the two opposite interior angles is 44°. Find the other opposite interior angle.

1. 106° − 44° = 62°

Answer: 62

Example

A 5-sided polygon has angles 105°, 102°, 112°, 106° and x°. Find x.

1. (5 − 2) × 180 = 540
2. 540 − 425 = 115°

Answer: 115

Example

What is each exterior angle of a regular polygon with 10 sides?

1. 360 ÷ 10 = 36°

Answer: 36

Example

A cylinder has radius 3 cm and height 3 cm. Find its volume in cm³, to 1 decimal place.

1. π × 3² × 3 = 27π
2. ≈ 84.8 cm³

Answer: 84.8

Example

A cone has radius 7 cm and height 18 cm. Its volume is kπ cm³. What is k?

1. ⅓ × 49 × 18 = 294
2. 294π cm³

Answer: 294

Example

A sphere has radius 3 cm. Its volume is kπ cm³. What is k?

1. ⁴⁄₃ × 27 = 36
2. 36π cm³

Answer: 36

Example

A cylinder has volume 117π cm³ and radius 3 cm. What is its height, in cm?

1. 117π ÷ (9π) = 13 cm

Answer: 13

Example

Find the total surface area of a closed cylinder with radius 1 cm and height 10 cm, to 1 decimal place.

1. 2 × π × 1² = 6.28
2. 2 × π × 1 × 10 = 62.83
3. ≈ 69.1 cm²

Answer: 69.1