Similar Triangles
In short: Similar triangles share the same shape but may differ in size: matching angles are equal and matching sides are all multiplied by one common scale factor.
GeometryGeometryG-SRT.B.5
What is Similar Triangles?
Similar triangles are copies made on a photocopier set to some percentage. Each angle is unchanged, while every side grows or shrinks by the same factor \(k\).
Writing \(\triangle ABC \sim \triangle DEF\) pairs the vertices in order, so \(\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC} = k\). Areas change by \(k^2\) while perimeters change by \(k\). The tool turns tall objects, such as trees and buildings, into proportions you can solve from a shadow or a mirror.
How to spot it
- Two triangles in a figure have the same angle marks but visibly different sizes.
- A shadow, mirror or ramp problem compares a short object with a tall one measured under the same sun.
- A line parallel to one side of a triangle cuts off a smaller triangle at the top.
- A proportion such as \(\frac{x}{12} = \frac{15}{9}\) is needed to find a missing side.
The method
- Show the triangles are similar by matching two angles, or by showing all three side ratios agree.
- List corresponding sides in matching order from the similarity statement.
- Write a proportion with one unknown and cross-multiply to solve.
- Check that the answer fits the scale factor: the bigger triangle’s sides must all be \(k\) times larger.
Similar Triangles: solved example
Problem
A 5.5-foot student casts a 3-foot shadow at the same moment a tree casts a 21-foot shadow. How tall is the tree?
Solution
The sun makes equal angles with the ground for both, and each object stands at a right angle to it, so the two shadow triangles are similar.
Set up the proportion of height to shadow: \(\frac{h}{21} = \frac{5.5}{3}\). Then \(h = \frac{5.5 \times 21}{3} = \frac{115.5}{3} = 38.5\). The tree is 38.5 feet tall.
Try it
Question 1
\(\triangle ABC \sim \triangle DEF\) with \(AB = 6\), \(BC = 9\), \(AC = 12\) and \(DE = 8\). Find \(EF\) and \(DF\).
Show the answer
The scale factor is \(\frac{8}{6} = \frac{4}{3}\). So \(EF = 9 \times \frac{4}{3} = 12\) and \(DF = 12 \times \frac{4}{3} = 16\).
Question 2
Two similar triangles have scale factor 3. If the small one has area 7 square units, what is the area of the large one?
Show the answer
Areas scale by \(k^2 = 9\), so the large area is \(9 \times 7 = 63\) square units.
Similar Triangles vs. Congruent Triangles
Congruent triangles are exact copies: the scale factor is 1, and matching sides are equal. These triangles only need matching angles and proportional sides, so sizes can differ. A 3-4-5 triangle and a 6-8-10 triangle are similar but not congruent; two 3-4-5 triangles are both.
Keep going
Lessons
Worksheets and quizzes
- Similar Triangle Theorems – Worksheet
- Similar Triangle Theorems – Quiz
- Dilations and Similarity – Worksheet
- Dilations and Similarity – Quiz
Related words
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