Key takeaway: Similar triangles have the same shape but not necessarily the same size: their corresponding angles are equal and their corresponding sides are in the same ratio.
Grade 10 – GeometryG-SRT.B.5Geometry and Measurement
Definition of Similar Triangles
Two triangles are similar triangles when one is an enlarged or shrunken copy of the other, possibly turned or flipped. Every pair of matching angles is equal, and every pair of matching sides has the same ratio, called the scale factor.
A photo and its wallet-size print show the idea: the shape is kept, the size changes. Similarity is the tool behind shadow measurements, map scales and many indirect-measurement problems, because a short measurable triangle can reveal an unmeasurable length.
Formula and notation
The symbol \(\sim\) means “is similar to.” If \(\triangle ABC \sim \triangle DEF\), then:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k \]
where \(k\) is the scale factor, and \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\).
Key facts
- The ratio of perimeters equals the scale factor k.
- The ratio of areas equals the square of the scale factor, k squared.
- Congruent triangles are the special case with a scale factor of 1.
- Two angle pairs are enough to prove similarity; see the AA rule.
Similar Triangles example
This is a similar triangles problem. At noon a 6-ft-tall hiker casts a 4.5-ft shadow, and a nearby flagpole casts a 27-ft shadow. How tall is the flagpole?
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The sun makes the same angle for both, so the height-and-shadow triangles are similar triangles and their sides are proportional.
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Set up: \(\frac{\text{pole}}{6} = \frac{27}{4.5}\).
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Scale factor: \(\frac{27}{4.5} = 6\), so the pole is \(6 \times 6 = 36\) ft.
The flagpole is 36 ft tall.
Common mistake
Watch out
Tomas notices the pole’s shadow is \(27 – 4.5 = 22.5\) ft longer than the hiker’s, so he adds that to the hiker’s height and answers \(6 + 22.5 = 28.5\) ft.
Why it is wrong
Similar figures grow by multiplying, not by adding. Every length is multiplied by the same scale factor, \(27 : 4.5 = 6\), so the height is \(6 \times 6 = 36\) ft. Adding differences keeps the gap but destroys the shape.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement