Triangle Similarity and Proportional Reasoning
Similar triangles have the same shape but different size; their corresponding sides are proportional.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
Set up a proportion using corresponding sides. The ratio of sides is the scale factor; the ratio of areas is that factor squared.
How to solve one, step by step
Example: similar triangles with ratio $2 : 3$; a side of $6$ corresponds to what?
- Set up the proportion: $\frac{2}{3} = \frac{6}{x}$.
- Cross-multiply: $2x = 18 \Rightarrow x = 9$.
The mistakes that cost points
- Pairing non-corresponding sides. Match sides that sit in the same position in each triangle.
- Confusing side ratio with area ratio. Areas scale by the square of the side ratio.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
Two similar triangles have corresponding sides in the ratio $3:5$. If the shorter triangle has a side of length $k$ corresponding to a side of length $20$ in the larger triangle, what is the value of $k$?
- A$\frac{20}{3}$
- B$\frac{9}{25} \times 20 = 7.2$
- C$15$
- D$12$
Show solution
Answer: D, $12$. $\frac{k}{20} = \frac{3}{5}$. So $k = \frac{3 \times 20}{5} = 12$.
Medium
Two similar triangles have areas in the ratio $16:25$. If the shorter triangle has a side of length $k$ and the corresponding side of the larger triangle has length $15$, what is the value of $k$?
- A$\frac{16}{25} \times 15 = 9.6$
- B$20$
- C$12$
- D$\frac{15 \times 16}{25} = 9.6$
Show solution
Answer: C, $12$. If the ratio of areas is $16:25$, then the ratio of corresponding sides is $\sqrt{\frac{16}{25}} = \frac{4}{5}$. So $\frac{k}{15} = \frac{4}{5}$, giving $k = 12$.
Hard
Two similar triangles have perimeters $24$ and $k$. The ratio of their corresponding sides is $3:4$. What is the value of $k$?
- A$36$
- B$32$
- C$\frac{9}{16} \times 24 = 13.5$
- D$18$
Show solution
Answer: B, $32$. Since the triangles are similar with side ratio $3:4$, all corresponding lengths (including perimeter) scale by the same ratio. So $\frac{24}{k} = \frac{3}{4}$, giving $k = \frac{24 \times 4}{3} = 32$.
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