Quiz: How well do you know the Perimeters and Areas of Similar Figures?

1. How does the ratio of the areas of two similar rectangles change if their sides are doubled?

Doubled

Quadrupled

Halved

No change

If the sides of two similar rectangles are doubled, their areas become quadrupled. This highlights the squared relationship between side lengths and areas in rectangles.

Explanation

If the sides of two similar rectangles are doubled, their areas become quadrupled. This highlights the squared relationship between side lengths and areas in rectangles.

2. If a rectangle's length and width are both doubled, how does the area change?

Doubled

Quadrupled

Halved

No change

Doubling both the length and width of a rectangle results in its area becoming quadrupled, showcasing the relationship between area and side length changes in similar shapes.

Explanation

Doubling both the length and width of a rectangle results in its area becoming quadrupled, showcasing the relationship between area and side length changes in similar shapes.

3. If a triangle's height is halved while its base remains unchanged, how does the area change?

Halved

Doubled

Quadrupled

No change

When the height of a triangle is halved while the base remains unchanged, its area becomes halved. This demonstrates the proportional relationship between height and area in similar triangles.

Explanation

When the height of a triangle is halved while the base remains unchanged, its area becomes halved. This demonstrates the proportional relationship between height and area in similar triangles.

4. What's the relationship between the areas of similar polygons?

Directly proportional

Inversely proportional

No relationship

Equal

The areas of similar polygons are directly proportional to the square of the ratio of their corresponding sides. This proportionality ensures that the areas grow or shrink consistently.

Explanation

The areas of similar polygons are directly proportional to the square of the ratio of their corresponding sides. This proportionality ensures that the areas grow or shrink consistently.

5. What's the ratio of the areas of two similar circles if their radii are in a 2:5 ratio?

2:5

4:25

5:2

25:4

The ratio of the areas of two similar circles is the square of the ratio of their radii. In this case, the ratio of radii is 2:5, leading to an area ratio of 4:25.

Explanation

The ratio of the areas of two similar circles is the square of the ratio of their radii. In this case, the ratio of radii is 2:5, leading to an area ratio of 4:25.

6. If the side length of a square is tripled, how does the area change?

Tripled

Nine times

Halved

No change

When the side length of a square is tripled, its area increases by a factor of nine. This emphasizes the squared relationship between side length and area in square-shaped similar figures.

Explanation

When the side length of a square is tripled, its area increases by a factor of nine. This emphasizes the squared relationship between side length and area in square-shaped similar figures.

7. What do similar figures have in common?

Same shape

Same perimeter

Same area

Same angles

Similar figures have the same shape but may differ in size. While angles remain congruent, lengths of corresponding sides are proportional, which leads to varying perimeters and areas.

Explanation

Similar figures have the same shape but may differ in size. While angles remain congruent, lengths of corresponding sides are proportional, which leads to varying perimeters and areas.

8. How does the area of a triangle change if its height and base are both doubled?

Doubled

Quadrupled

Halved

No change

Doubling both the height and base of a triangle leads to a quadrupled area. This relationship highlights the direct relationship between the length of the base and the height of a triangle.

Explanation

Doubling both the height and base of a triangle leads to a quadrupled area. This relationship highlights the direct relationship between the length of the base and the height of a triangle.

9. What's true about the angles in similar figures?

They are congruent.

They are acute.

They are obtuse.

They vary.

The angles in similar figures remain congruent. Regardless of size changes, corresponding angles maintain their measures, contributing to the overall similarity of the figures.

Explanation

The angles in similar figures remain congruent. Regardless of size changes, corresponding angles maintain their measures, contributing to the overall similarity of the figures.

10. How are the areas of similar triangles related if the ratio of their sides is k:1?

K^2 times

K times

K^3 times

K−1 times

The areas of similar triangles are proportional to the square of the ratio of their corresponding sides. This relationship is vital for solving problems involving similar triangles and their areas.

Explanation

The areas of similar triangles are proportional to the square of the ratio of their corresponding sides. This relationship is vital for solving problems involving similar triangles and their areas.

11. What's the relationship between the perimeters of similar polygons?

Directly proportional

Inversely proportional

No relationship

Equal

The perimeters of similar polygons are directly proportional to their corresponding sides. Doubling the side lengths results in a doubled perimeter, maintaining this linear relationship.

Explanation

The perimeters of similar polygons are directly proportional to their corresponding sides. Doubling the side lengths results in a doubled perimeter, maintaining this linear relationship.

12. If one side of a triangle is proportionally doubled, how does the area change?

Doubled

Quadrupled

Halved

No change

When a side of a triangle is doubled, its area becomes quadrupled due to the relationship between sides and areas in similar figures.

Explanation

When a side of a triangle is doubled, its area becomes quadrupled due to the relationship between sides and areas in similar figures.

13. What's the ratio of the perimeters of two similar rectangles if their sides are in a 3:4 ratio?

3:4

4:3

9:16

16:9

The ratio of the perimeters of two similar rectangles is equal to the ratio of their corresponding sides. In this case, the ratio is 3:4, leading to a perimeter ratio of 3:4

Explanation

The ratio of the perimeters of two similar rectangles is equal to the ratio of their corresponding sides. In this case, the ratio is 3:4, leading to a perimeter ratio of 3:4

14. How does the ratio of the perimeters of two similar squares change if their sides are doubled?

Doubled

Quadrupled

Halved

No change

Doubling the sides of two similar squares results in a doubled perimeter ratio. This linear relationship between side lengths and perimeters remains consistent in square-shaped similar figures.

Explanation

Doubling the sides of two similar squares results in a doubled perimeter ratio. This linear relationship between side lengths and perimeters remains consistent in square-shaped similar figures.

15. What's the ratio of the areas of two similar squares if their side lengths are in a 1:3 ratio?

1:3

1:9

3:1

9:1

The ratio of the areas of two similar squares is the square of the ratio of their side lengths. In this case, the ratio of side lengths is 1:3, resulting in an area ratio of 1:9.

Explanation

The ratio of the areas of two similar squares is the square of the ratio of their side lengths. In this case, the ratio of side lengths is 1:3, resulting in an area ratio of 1:9.