Math / Geometry

QuestionIn the diagram below, the two triangles are similar. Determine the values of $x$ and $y$ .

Studdy Solution

STEP 1

1. Triangles $\triangle ABC$ and $\triangle ADE$ are similar.
2. Corresponding sides of similar triangles are proportional.

STEP 2

1. Set up the proportion using corresponding sides.
2. Solve for $x$ .
3. Solve for $y$ .

STEP 3

Set up the proportion using corresponding sides of the similar triangles:
Since $\triangle ABC \sim \triangle ADE$ , we have:
$\frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE}$
Substitute the known values:
$\frac{7.5}{5} = \frac{15}{x} = \frac{AC}{6}$

STEP 4

Solve for $x$ using the proportion:
From $\frac{7.5}{5} = \frac{15}{x}$ :
$\frac{7.5}{5} = \frac{15}{x}$
Cross-multiply to solve for $x$ :
$7.5x = 5 \times 15$
$7.5x = 75$
$x = \frac{75}{7.5}$
$x = 10$

STEP 5

Solve for $y$ using the proportion:
From $\frac{7.5}{5} = \frac{AC}{6}$ :
$\frac{7.5}{5} = \frac{AC}{6}$
Cross-multiply to solve for $AC$ :
$7.5 \times 6 = 5 \times AC$
$45 = 5 \times AC$
$AC = \frac{45}{5}$
$AC = 9$
Thus, $y = 9$ .
The values are:
$x = 10$ $y = 9$

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