SAT similar triangles: set up the proportion and the rest is arithmetic
Similar triangles have the same shape at different sizes, so their sides share one ratio. Learn to match corresponding sides, use the parallel-line shortcut, and solve shadow problems.
SPSAT Practice Editorial7 min read
The short version
What you'll take away
- Two triangles with two pairs of equal angles are similar, and all their corresponding sides share one ratio.
- Match sides by the angles they are opposite, not by their position on the page.
- A line parallel to one side of a triangle cuts off a smaller, similar triangle.
In this article
Same shape, one ratio
Similar triangles have equal corresponding angles, and their corresponding sides are proportional. If one triangle is twice as large, every side is twice as long. Two pairs of equal angles are enough to prove similarity, because the third angles must then match too.
See the skill
A 3-4-5 triangle and a scaled copy
4/8 = 3/6 = 5/x → x = 10
Match sides by angles
Diagrams on the SAT are not always drawn in the same orientation. Match each side by the angle across from it: the side opposite the smallest angle in one triangle corresponds to the side opposite the smallest angle in the other. The shortest side matches the shortest side, and the longest matches the longest.
Then set up one proportion with the unknown and a known pair: \(\displaystyle \frac{\text{small side}}{\text{large side}} = \frac{\text{small side}}{\text{large side}}\), keeping the same triangle on top on both sides.
The parallel-line shortcut
If segment DE is drawn inside triangle ABC parallel to BC, then triangle ADE is similar to triangle ABC, because they share angle A and the parallel lines create equal corresponding angles. If AD = 4, DB = 6, and DE = 5, then AB = 10 and \(\displaystyle \frac{AD}{AB} = \frac{DE}{BC}\), so \(\displaystyle \frac{4}{10} = \frac{5}{BC}\) and BC = 12.5.
Shadows and other real-world triangles
At the same time of day, the sun’s rays hit objects at the same angle, so a person and their shadow form a triangle similar to a tree and its shadow. Heights and shadows are proportional.
Your turn
Quick practice
A 6-foot-tall person casts a 4-foot shadow. At the same time, a tree casts a 30-foot shadow. How tall is the tree, in feet?
Hint: Set height over shadow for each object.
Practice: try these before checking the answers
Match corresponding sides first.
- 1. A triangle has sides 5, 7, and 9. In a similar triangle, the side matching 5 is 15. Find the other two sides.
- 2. In triangle ABC, DE ∥ BC with D on AB. AD = 3, AB = 9, and DE = 4. Find BC.
- 3. Two similar triangles have a scale factor of 3. What is the ratio of their areas?
- 4. A right triangle has legs 6 and 8. A similar triangle has hypotenuse 25. Find its legs.
Sources & further reading
One step from here
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