Similar Figures
In short: Similar figures share one shape but not always one size: their corresponding angles match exactly, and their corresponding sides are all stretched or shrunk by the same factor.
8th GradeGeometry8.G.A.4
What is Similar Figures?
A photo and its poster-size print show similar figures: nothing is distorted, the picture is only larger. A dilation, perhaps followed by a rotation, a reflection or a slide, carries one onto the other.
In symbols, the angles in matching corners are congruent, and each length in the new figure equals \(k\) times its partner in the old one. The constant \(k\) is the scale factor; when \(k = 1\) the pair is even congruent.
How to spot it
- A problem mentions a model, a map, a photo or a shadow and asks for a real length.
- Two shapes look alike, with equal angles but sides of different lengths.
- You are given a pair of matching sides and asked for a missing side in the other shape.
- The wording says dilation, enlargement, reduction or scale factor.
The method
- List the matching sides in the same order, checking that equal angles sit in matching corners.
- Divide one side of the new figure by its partner in the original to get the scale factor \(k\).
- Check that a second pair of sides gives the same ratio; if not, the figures are not similar.
- Multiply a known side by \(k\) to find a missing side, or divide by \(k\) to go back.
Similar Figures: solved example
Problem
At 4:00 p.m. in Tulsa, Marisol is 5.5 feet tall and her shadow measures 4.4 feet. A flagpole beside her has a shadow of 17.6 feet. What is the height of the flagpole?
Solution
The sun’s rays make the same angle with the ground, so the student with her shadow and the pole with its shadow form two right triangles with equal angles. They are similar.
Scale factor from shadow to shadow: \(\frac{17.6}{4.4} = 4\). The pole is 4 times as tall as the student: \(5.5 \times 4 = 22\). The flagpole is 22 feet tall.
Try it
Question 1
One rectangle is 7 cm long and 4 cm wide. A similar, larger rectangle is 17.5 cm long. Find its width.
Show the answer
\(k = \frac{17.5}{7} = 2.5\), so the width is \(4 \times 2.5 = 10\) cm.
Question 2
Could a 6 by 9 card and a 10 by 14 poster be similar?
Show the answer
\(\frac{10}{6} \approx 1.67\) but \(\frac{14}{9} \approx 1.56\). The ratios differ, so they are not similar.
Similar Figures vs. Congruent Figures
Congruent figures are exact copies: same shape and same size, so the scale factor is 1. Similar figures only need the same shape. Every congruent pair is similar, but a pair with scale factor 3 is similar and certainly not congruent.
Keep going
Lessons
Worksheets and quizzes
- Dilations and Similar Figures – Worksheet
- Dilations and Similar Figures – Quiz
- Slope and Similar Triangles – Worksheet
- Slope and Similar Triangles – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Geometry