Question

Determine the length of z.

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asked 2021-08-05
Determine the length of z.
Given:
image

Answers (1)

2021-08-06
In \(\displaystyle\triangle{A}{B}{C}\ {\quad\text{and}\quad}\ \triangle{A}{D}{C}\),
\(\displaystyle\angle{C}{A}{B}\stackrel{\sim}{=}\angle{C}{D}{A}\)
\(\displaystyle\angle{A}{C}{D}\stackrel{\sim}{=}\angle{A}{C}{B}\)
So, by AA Similarity Postulate, \(\displaystyle\triangle{C}{A}{B}\sim\triangle{C}{D}{A}\).
Hence,
\(\displaystyle{\frac{{{A}{D}}}{{{A}{B}}}}={\frac{{{D}{C}}}{{{A}{C}}}}={\frac{{{A}{C}}}{{{B}{C}}}}\)
\(\displaystyle\Rightarrow{\frac{{{x}}}{{{y}}}}={\frac{{{12.5}}}{{{z}}}}={\frac{{{z}}}{{{20.5}}}}\)
\(\displaystyle\Rightarrow{\frac{{{12.5}}}{{{z}}}}={\frac{{{z}}}{{{20.5}}}}\)
\(\displaystyle\Rightarrow{z}^{{{2}}}={\left({12.5}\right)}{\left({20.5}\right)}\)
\(\displaystyle\Rightarrow{z}^{{{2}}}={256.25}\)
\(\displaystyle\Rightarrow{z}\approx{16}\)
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