A free printable worksheet that pairs with the Similarity Builder game. The worksheet prints first, and the answer key prints on its own page after it, so you can leave it off. To make a PDF, click Print and choose "Save as PDF."

Similar Figures and Scale Factor

Grades 7–8 · CCSS 7.G.A.1, 8.G.A.4 · Pairs with the Similarity Builder game

Name

Date

Similar figures have equal angles and proportional sides. Scale factor k = image side ÷ original side. k > 1 is an enlargement; k < 1 is a reduction. Image side = k × original side. Always match sides that are in the same position.

Part A: Find the scale factor (from the first figure to the second)

1. Two similar rectangles.

k =

2. A triangle with sides 3, 4 and 5 is enlarged to sides 9, 12 and 15.

k =

3. A triangle with sides 8, 10 and 12 is shrunk to sides 4, 5 and 6.

k =

4. A square with side 3 is dilated to a square with side 12.

k =

Part B: Use the scale factor

5. k = 4. The original side is 6. Find the image side.

6. k = 3. The image side is 21. Find the original side.

7. k = ½. The original side is 18. Find the image side.

8. k = 1.5. The original side is 8. Find the image side.

Part C: Find k first, then the missing side

9. The rectangles are similar. Find the missing side.

? =

10. A small triangle has sides 4, 6 and 7. In a similar big triangle, the side that matches 4 is 10. Find the other two sides of the big triangle.

k = sides ,

Part D: Similar triangles in the real world

11. A line parallel to the base of a triangle cuts off a small triangle with base 4. The big triangle's base is 10. A side of the small triangle is 6. Find the matching side of the big triangle.

12. A 5 ft student casts a 3 ft shadow. At the same time, a tree casts an 18 ft shadow. How tall is the tree?

Part E: Similar or not?

13. Are a 2 × 3 rectangle and a 4 × 6 rectangle similar? Are a 2 × 3 rectangle and a 4 × 5 rectangle similar? Explain both.

14. Find the mistake. Casey said: "3 grew to 9, which is 6 more, so the side of 4 grows to 10." What went wrong? What's the right answer?

Correct:

Answer Key: Similar Figures and Scale Factor

For the teacher

10 / 4 = 15 / 6 → k = 2.5
9 / 3 → k = 3
4 / 8 → k = ½ (a reduction)
12 / 3 → k = 4
6 × 4 = 24
21 ÷ 3 = 7
18 × ½ = 9
8 × 1.5 = 12
k = 15 / 5 = 3; 8 × 3 = 24
k = 10 / 4 = 2.5; 6 × 2.5 = 15 and 7 × 2.5 = 17.5
k = 10 / 4 = 2.5; 6 × 2.5 = 15
Height : shadow is the same for both. k = 18 / 3 = 6; 5 × 6 = 30 ft
2 × 3 and 4 × 6: similar (both sides × 2). 2 × 3 and 4 × 5: not similar (4/2 = 2 but 5/3 ≠ 2).
Casey added 6 instead of multiplying. The scale factor is 9 / 3 = 3, so 4 × 3 = 12

In the game, the answer is the size the bridge is built at, so an additive answer like Casey's builds a bridge that's the wrong size and fails. Matching sides share a color in the game, which helps with turned figures.