Math / Geometry

Question18 pts
6. $\triangle \mathrm{ABC} \sim \triangle \mathrm{ADE}$ $\begin{array}{l} x= \\ r= \end{array}$

Studdy Solution

STEP 1

1. $\triangle \mathrm{ABC} \sim \triangle \mathrm{ADE}$ indicates that the triangles are similar.
2. Similar triangles have corresponding sides in proportion.
3. We are given the lengths of sides $\overline{AB}$ , $\overline{AC}$ , $\overline{BC}$ , and $\overline{DE}$ .
4. We need to find the lengths of $\overline{AE} = x$ and $\overline{AD} = y$ .

STEP 2

1. Use the properties of similar triangles to set up proportions.
2. Solve the proportion to find $x$ .
3. Solve the proportion to find $y$ .

STEP 3

Since $\triangle \mathrm{ABC} \sim \triangle \mathrm{ADE}$ , the corresponding sides are proportional. This means:
$\frac{\overline{AB}}{\overline{AD}} = \frac{\overline{AC}}{\overline{AE}} = \frac{\overline{BC}}{\overline{DE}}$
Substitute the known values:
$\frac{3}{y} = \frac{4}{x} = \frac{5}{12.5}$

STEP 4

First, solve for $x$ using the proportion $\frac{4}{x} = \frac{5}{12.5}$ .
Cross-multiply to solve for $x$ :
$4 \cdot 12.5 = 5 \cdot x$
$50 = 5x$
Divide both sides by 5:
$x = 10$

STEP 5

Next, solve for $y$ using the proportion $\frac{3}{y} = \frac{5}{12.5}$ .
Cross-multiply to solve for $y$ :
$3 \cdot 12.5 = 5 \cdot y$
$37.5 = 5y$
Divide both sides by 5:
$y = 7.5$
The values are:
$x = 10$ $y = 7.5$

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