Similar triangles have the same shape but different sizes. Their corresponding angles are equal and their corresponding sides are proportional. The SAT tests this concept in both direct geometry and word problem formats.

What Makes Triangles Similar?

Two triangles are similar if:

On the SAT, AA is the most common test. If two angles match, the triangles are similar.

Setting Up Proportions

Once you know triangles are similar, set up a proportion:

Example 1: Triangle ABC ~ Triangle DEF. If , , , find .




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Parallel Lines Create Similar Triangles

When a line is parallel to one side of a triangle and cuts the other two sides, it creates a smaller similar triangle.

Example 2: In triangle ABC, line DE is parallel to BC, where D is on AB and E is on AC. If , , and , find .

The scale factor is

The Shadow Problem

A classic SAT application:

Example 3: A 6-foot person casts a 4-foot shadow. At the same time, a tree casts a 20-foot shadow. How tall is the tree?

The sun creates the same angle for both, so the triangles are similar:


Scale Factor, Area, and Volume

If the scale factor between similar figures is :

Example 4: Two similar triangles have sides in a ratio of . If the smaller triangle has area 18, what is the area of the larger?

Practice Problems

Problem 1: Triangle PQR ~ Triangle XYZ. , , . Find .

Solution

Problem 2: A flagpole casts a 12-foot shadow while a 5-foot stick casts a 3-foot shadow. How tall is the flagpole?

Solution

feet

Problem 3: Two similar rectangles have a length ratio of . The smaller has area 12. What is the area of the larger?

Solution

Area ratio = . So .

Key Takeaways

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