In right triangle XYZ, the measure of angle XYZ is 90^\circ. Altitude YW is drawn to hypot… — SAT

In right triangle $XYZ$ , the measure of angle $XYZ$ is $90^\circ$ . Altitude $YW$ is drawn to hypotenuse $XZ$ , dividing the triangle into two smaller triangles, $\triangle XYW$ and $\triangle YZW$ . If the area of $\triangle XYW$ is $9$ and the area of $\triangle YZW$ is $36$ , what is the length of altitude $YW$ ?

A
$3\sqrt{2}$
$6$ Answer
C
$12$
D
$18$

Answer

The length of altitude

YW

6

Because altitude

YW

is perpendicular to hypotenuse

XZ

in right triangle

XYZ

, the two smaller triangles

\triangle XYW

and

\triangle YZW

are similar. The ratio of the area of

\triangle XYW

to the area of

\triangle YZW

9

36

, which simplifies to

1

4

. Since the ratio of the areas of similar triangles is the square of their linear scale factor, the ratio of their corresponding side lengths is

\sqrt{1/4} = 1/2

. Side

XW

corresponds to side

YW

, and side

YW

corresponds to side

WZ

, so

YW = 2 \cdot XW

. The area of

\triangle XYW

is given by

\frac{1}{2} \cdot XW \cdot YW = 9

. Substituting

YW = 2 \cdot XW

gives

\frac{1}{2} \cdot XW \cdot (2 \cdot XW) = 9

, which simplifies to

XW^2 = 9

, so

XW = 3

. Thus, the length of altitude

YW

2 \cdot 3 = 6

Step-by-Step Solution

Set up the similarity relationship between the two smaller triangles.

\triangle XYW \sim \triangle YZW

because both are right triangles sharing an acute angle relationship with the main right triangle

\triangle XYZ

An altitude drawn to the hypotenuse of a right triangle divides it into two triangles that are similar to the original triangle and to each other.

Determine the scale factor between the similar triangles using their areas.

The ratio of the area of

\triangle XYW

to the area of

\triangle YZW

9/36 = 1/4

. The linear scale factor is the square root of the area ratio, which is

\sqrt{1/4} = 1/2

For similar figures, the ratio of their areas is the square of the ratio of their corresponding linear dimensions.

Express the relationship between the corresponding sides of the two triangles.

\triangle XYW

and

\triangle YZW

, the side

XW

corresponds to

YW

, and the side

YW

corresponds to

WZ

. Therefore,

YW = 2 \cdot XW

The linear scale factor of

1/2

means each side of the smaller triangle is half the length of the corresponding side of the larger triangle.

Use the area formula for

\triangle XYW

to solve for the length of

XW

\text{Area}(\triangle XYW) = \frac{1}{2} \cdot XW \cdot YW = 9 \implies \frac{1}{2} \cdot XW \cdot (2 \cdot XW) = 9 \implies XW^2 = 9 \implies XW = 3

Substituting the relationship between

YW

and

XW

into the area equation isolates the single variable

XW

Calculate the length of the altitude

YW

YW = 2 \cdot 3 = 6

Multiplying the length of

XW

by the scale factor of

2

yields the length of

YW

Key Concept

Similarity in right triangles and the geometric relationships of altitudes

Estimated Time:2m 30s

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