Questions: Triangle ABC and triangle XYZ are similar. Find the missing side length. (The triangles are not drawn to scale.)

Triangle ABC and triangle XYZ are similar. Find the missing side length. (The triangles are not drawn to scale.)
Transcript text: $\triangle A B C$ and $\triangle X Y Z$ are similar. Find the missing side length. (The triangles are not drawn to scale.)
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Solution

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Solution Steps

Step 1: Identify the Corresponding Sides

Since triangles \( \triangle ABC \) and \( \triangle XYZ \) are similar, their corresponding sides are proportional. Identify the corresponding sides:

  • \( AB \) corresponds to \( XY \)
  • \( BC \) corresponds to \( YZ \)
  • \( AC \) corresponds to \( XZ \)
Step 2: Set Up the Proportion

Given:

  • \( AB = 6 \)
  • \( AC = 5 \)
  • \( BC = 8 \)
  • \( XY = 18 \)
  • \( XZ = 15 \)

We need to find \( YZ \). Set up the proportion using the corresponding sides: \[ \frac{BC}{YZ} = \frac{AB}{XY} \]

Step 3: Solve the Proportion

Substitute the known values into the proportion: \[ \frac{8}{YZ} = \frac{6}{18} \]

Simplify the right side of the equation: \[ \frac{6}{18} = \frac{1}{3} \]

So the proportion becomes: \[ \frac{8}{YZ} = \frac{1}{3} \]

Cross-multiply to solve for \( YZ \): \[ 8 \cdot 3 = 1 \cdot YZ \] \[ 24 = YZ \]

Final Answer

\[ YZ = 24 \]

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