Teacher answer key · Grades 7–10 · about 20 minutes

Answer key: Similarity & Scale Factor Worksheet

For teachers. This page is hidden from search engines and not linked from student pages, but it is not password-protected, so please don’t share the link with students. Student worksheet: Similarity & Scale Factor Worksheet.

Goal: Dilate a triangle by different scale factors, and discover how lengths, areas and volumes change with the scale factor.

Connects to Common Core 7.G.A.1 (Solve problems involving scale drawings, including computing actual lengths and areas from a scale drawing.); HSG-SRT.A.1 (Verify experimentally the properties of dilations given by a center and a scale factor.); 8.G.A.3 (Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.)

Completed results table

Values are the readouts of the tool (Transformation playground) for each setting.

Setting	A′	A′B′ length	New area	Area factor
Scale factor 0.5	(-1, -0.5)	2	1.5 units²	0.25×
Scale factor 1	(-2, -1)	4	6 units²	1×
Scale factor 1.5	(-3, -1.5)	6	13.5 units²	2.25×
Scale factor 2	(-4, -2)	8	24 units²	4×
Scale factor 3	(-6, -3)	12	54 units²	9×

Answers

The original side AB is 4 units long. How does the length of A′B′ compare with the scale factor?
A′B′ = 4 × k: it is 2 for k = 0.5, 8 for k = 2 and 12 for k = 3. Every length is multiplied by the scale factor.
Look at the area factor column. How is it related to the scale factor?
The area factor is k²: 0.5² = 0.25, 1.5² = 2.25, 2² = 4 and 3² = 9. Area is length × length, and both lengths are multiplied by k.
Predict the new area for a scale factor of 2.5, then check it with the tool.
New area = 6 × 2.5² = 6 × 6.25 = 37.5 square units.
A map has a scale of 1 : 50,000. Two towns are 6 cm apart on the map. How far apart are they in real life? A park covers 2 cm² on the map: what is its real area?
6 cm × 50,000 = 300,000 cm = 3 km. Areas scale by 50,000², so 2 cm² × 2,500,000,000 = 5,000,000,000 cm² = 0.5 km² (1 km² = 10,000,000,000 cm²).
A cube with 3 cm edges is scaled by a factor of 2. Find the new surface area and volume, and compare the growth with k² and k³.
The new edge is 6 cm. The surface area grows from 54 to 216 cm² (× 4 = 2²), and the volume grows from 27 to 216 cm³ (× 8 = 2³).
Is a dilation with scale factor 1 a similarity? With scale factor 0.5, is the image similar to the original?
Yes to both. Scale factor 1 gives an identical (congruent) triangle, and congruent figures are similar with a ratio of 1. With 0.5, every side is halved and every angle stays the same, so the image is similar, just smaller, with an area factor of 0.25.

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