﻿ Proportions and Similar Figures Worksheet | Problems & Solutions

# Proportions and Similar Figures Worksheet

Proportions and Similar Figures Worksheet
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1.
The area of a square is 32 m2. Find the area of a similar square whose dimensions are quadrupled.
 a. 576 m2 b. 484 m2 c. 512 m2 d. 476 m2

#### Solution:

If the ratio of corresponding edge lengths of two similar polygons is a : b, then the ratio of their areas is a2 : b2.

The ratio of area of the squares = 12 : 42 = 1 : 16

Area of smaller squareArea of bigger square = 116

Therefore, area of bigger square = 16 × area of smaller square = 16 × 32 = 512 m2.
[Substitute and simplify.]

2.
Linda plays in a square-shaped ground beside their house, which has a side length of 24 m. She asks her dad to make a similar playing area in their house, whose area is $\frac{1}{16}$ of the area of the ground. What must be the side length of the new ground?
 a. 15 m b. 6 m c. 12 m d. 3 m

#### Solution:

If the ratio of corresponding edge lengths of two similar polygons is a : b, then the ratio of their areas is a2 : b2

The ratio of areas of the grounds is 1 : 16 = 12 : 42

Length of the smaller square groundLength of the bigger square ground = 14

So, the ratio of the lengths of the grounds is 1 : 4

Therefore, length of the smaller square ground = 14 × length of bigger square ground = 244 = 6 m
[Substitute and simplify.]

3.
The measure of each angle of a regular hexagon is 120°. A similar hexagon is formed by tripling the dimensions of the original hexagon. What will be the measure of each angle of the new hexagon?
 a. 240° b. 120° c. 108° d. 60°

#### Solution:

If two polygons are similar, then corresponding angles are congruent.

Therefore, the measure of each angle of the new hexagon is 120°.

4.
A regular octagon measures 10 cm on each side. A similar octagon is formed by halving the side length of the original octagon. What will happen to the perimeter of the new octagon?
 a. perimeter does not change b. perimeter triples c. perimeter doubles d. perimeter halves

#### Solution:

Side length of the original octagon = 10 cm

Perimeter of the original octagon = 8 × 10 = 80 cm
[Perimeter of an octagon = Number of sides × Length of each side.]

Side length of the new octagon = 10 / 2 = 5 cm
[Side length of the new octagon is halved.]

Perimeter of the new octagon = 8 × 5 = 40 cm

= 1 / 2(80) = 1 / 2(Perimeter of the original octagon)

So, perimeter of the new octagon is half the perimeter of the original octagon.

5.
The ratio of the areas of two regular hexagons is 1 : 36. Find the perimeter of the smaller hexagon, if the side length of the bigger hexagon is 12 cm.
 a. 6 cm b. 36 cm c. 1 cm d. 12 cm

#### Solution:

If the ratio of corresponding areas of two similar polygons is a2 : b2, then the ratio of their edge lengths is a : b.

The ratio of the areas is 1 : 36 = 12 : 62.

Length of the smaller hexagonLength of the bigger hexagon = 16

Length of the smaller hexagon = Length of the bigger hexagon × 1 / 6

So, the length of the smaller hexagon = 12 × 1 / 6 = 2 cm.
[Substitute and simplify.]

Perimeter of the smaller hexagon = Length of the smaller hexagon × Number of sides of a hexagon = 2 × 6 = 12 cm
[Substitute and simplify.]

So, the perimeter of the smaller hexagon is 12 cm.

6.
The radii of two similar spheres are 12 cm and 18 cm. Find the ratio of their volumes.
 a. 8 : 27 b. 3 : 4 c. 27 : 8 d. 2 : 3

#### Solution:

The radii of the two similar spheres are a = 12 cm and b = 18 cm.
[Given.]

[Formula.]

= a / b = 12 / 18= 2 / 3

The ratio of volumes of the spheres is a3 : b3 = 23 : 33 = 8 : 27.

7.
Josh wants to make an advertisement board similar to another advertisement board, which has a length and width of 18 feet and 12 feet. He wants to have a length of 12 feet for the new board. What will be the width of the board that he wants to make?
 a. 10 feet b. 8 feet c. 9 feet d. 7 feet

#### Solution:

Let n be the required width of the board.

1218 = n12
[Write a proportion.]

n × 18 = 12 × 12
[Write the cross products.]

n = 8
[Divide both sides by 18.]

The width of the board that Josh wants to make, is 8 feet.

8.
Sam saw a house and built a model similar to the house. The height and width of the model are in the ratio of 1 : 3. If the width of the model is 9 feet, then what is the height of the model?
 a. 2 feet b. 4 feet c. 5 feet d. 3 feet

#### Solution:

Let n be the height of the model.

13 = n9
[Write a proportion]

1 × 9 = 3 × n
[Write a cross product.]

3 = n
[Simplify.]

The height of the model is 3 feet.

9.
A regular hexagon measures 9 cm on each side. A similar hexagon is formed by halving the side length of the original one. What will happen to perimeter of the new hexagon?
 a. becomes four times the perimeter of the original hexagon b. becomes half the perimeter of the original hexagon c. becomes two times the perimeter of the original hexagon d. perimeter does not change

#### Solution:

Side length of the original hexagon = 9 cm

Perimeter of the original hexagon = 6 × 9 = 54 cm
[Perimeter of an hexagon = Number of sides × Length of each side.]

Side length of the new hexagon = 9 / 2 = 4.5 cm
[Side length of the new hexagon is halved.]

Perimeter of the new hexagon = 6 × 4.5 = 27 cm

= 1 / 2(54) = 1 / 2(Perimeter of the original hexagon)

So, perimeter of the new hexagon becomes half the perimeter of the original hexagon.

10.
The side length of a heptagon is given as 10 in. A similar heptagon is formed by multiplying the side length of the original one by 3. What will happen to perimeter of the new heptagon?

 a. perimeter does not change b. becomes two times the perimeter of the original heptagon c. becomes three times the perimeter of the original heptagon d. becomes four times the perimeter of the original heptagon

#### Solution:

Side length of the original heptagon = 10 in.

Perimeter of the original heptagon = 7 × 10 = 70 in.
[Perimeter of an heptagon = Number of sides × Length of each side.]

Side length of the new heptagon = 3 ×10 = 30 in.
[Side length of the new heptagon is multiplied by 3.]

Perimeter of the new heptagon = 7 × 30 = 210 in.

= 3(70) = 3(Perimeter of the original heptagon)

So, perimeter of the new heptagon becomes three times the perimeter of the original heptagon.