Questions: Given that triangle ABC is similar to triangle A'B'C', solve for x and y. Use fractions as your answer where helpful. Round to three decimal places.

$Given that triangle ABC is similar to triangle A'B'C', solve for x and y. Use fractions as your answer where helpful. Round to three decimal places.$

Transcript text: Given that $\triangle A B C \sim \Delta A^{\prime} B^{\prime} C^{\prime}$, solve for $x$ and $y$. Use fractions as your answer where helpful. Round to three decimal places

Solution

Solution Steps

Step 1: Identify the relationship between the triangles

Given that $\triangle ABC \sim \triangle A'B'C'$, the corresponding sides of the triangles are proportional. This means: \[ \frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{CA}{C'A'} \]

Step 2: Set up the proportion for $x$

Using the corresponding sides $AB$ and $A'B'$: \[ \frac{30 + x}{22} = \frac{30}{14} \]

Step 3: Solve for $x$

Cross-multiply to solve for $x$: \[ (30 + x) \cdot 14 = 30 \cdot 22 \] \[ 420 + 14x = 660 \] \[ 14x = 240 \] \[ x = \frac{240}{14} = 17.143 \]

Step 4: Set up the proportion for $y$

Using the corresponding sides $BC$ and $B'C'$: \[ \frac{y + 15}{30} = \frac{y}{14} \]

Step 5: Solve for $y$

Cross-multiply to solve for $y$: \[ (y + 15) \cdot 14 = y \cdot 30 \] \[ 14y + 210 = 30y \] \[ 210 = 16y \] \[ y = \frac{210}{16} = 13.125 \]