Before you read

Answer first — see what you already know.

A scale model of a bridge uses similar triangles. In the model, $PQ = 4$ cm and the corresponding real measurement $ST = 10$ m. If $QR = 6$ cm in the model, what is the real length $T U$ ?

A
10
B
12
C
15
D
24

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A 6-foot surveyor casts a 4-foot shadow. A nearby communications tower casts a 20-foot shadow at the same time. How tall is the tower?

A
20 ft
B
24 ft
C
30 ft
D
32 ft

Check your answer

Answer:

Theory

Similar Triangles

Similar triangles ( $\sim$ ): Same shape, possibly different sizes.
- All corresponding angles are equal
- All corresponding sides are proportional: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k$ (scale factor)

Original triangle

Similar triangle (scale factor 2)

Tests for similarity: AA (two angles equal), SSS (all sides proportional), SAS (two sides proportional + included angle equal)

Congruent triangles ( $\cong$ ): Same shape AND same size ( $k = 1$ ).

Key fact: If two triangles are similar with scale factor $k$ , then the ratio of their areas is $k^2$ .

Example 1

Triangle $ABC \sim DEF$ . If $AB = 6$ , $DE = 9$ , and $BC = 8$ , find $EF$ .

Scale factor: $\frac{DE}{AB} = \frac{9}{6} = 1.5$

$EF = BC \times 1.5 = 8 \times 1.5 = 12$

EF = 12

Strategy

SAT Pro Tip

On the SAT, similar triangles often appear when: (1) two triangles share an angle and have parallel sides, (2) a line parallel to one side of a triangle divides the other two sides proportionally, or (3) right triangles share an angle.

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