"Determine the length of x. Given:" was answered by our experts

Suman Cole

Answered question

2021-08-03

Determine the length of x.
Given:

Answer & Explanation

d2saint0

Skilled2021-08-04Added 89 answers

In

△ A B C and △ A D C

,

∠ C A B \tilde{=} ∠ C D A

∠ A C D \tilde{=} ∠ A C B

So, by AA Similarity Postulate,

△ C A B \sim △ C D A

.
Hence,

\frac{A D}{A B} = \frac{D C}{A C} = \frac{A C}{B C}

\Rightarrow \frac{x}{y} = \frac{12.5}{z} = \frac{z}{20.5}

\Rightarrow \frac{12.5}{z} = \frac{z}{20.5}

\Rightarrow z^{2} = (12.5) (20.5)

\Rightarrow z^{2} = 256.25

\Rightarrow z \approx 16

Similarity,

△ B A C \sim △ B D A

Hence,

\frac{A D}{C A} = \frac{B D}{B A} = \frac{B A}{B C}

\Rightarrow \frac{x}{z} = \frac{8}{y} = \frac{y}{20.5}

\Rightarrow \frac{8}{y} = \frac{y}{20.5}

\Rightarrow \frac{8}{y} = \frac{y}{20.5}

\Rightarrow y^{2} = 8 (20.5)

\Rightarrow y^{2} = 164

\Rightarrow y \approx 12.8

Since,

\frac{x}{z} = \frac{8}{y}

\Rightarrow x = z \frac{8}{y}

\Rightarrow x \approx 16 \frac{8}{12.8}

\Rightarrow x = 10

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