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Calculating Heights and Distances

In the given ...

Question

In the given figure, $∠ A C E = 30^{o}, ∠ A D B = 45^{o}, C E = 12 m$ and $C D = 10 m .$ Find $A D$ .

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Solution

$C E = 12 m = D B (s i n c e B D C E i s r e c t a n g l e)$
$C D = 10 m = B E$
In triangle $A B D$
$tan 45^{o} = (\frac{A B}{B D})$
$1 = \frac{A B}{B D} \Rightarrow A B = B D = 12 m$
${A D}^{2} = {A B}^{2} + {B D}^{2} (A B D i s a r i g h t t r i a n g l e)$
${A D}^{2} = 12^{2} + 12^{2} = 288$
$\Rightarrow A D = 12 \sqrt{2} m$

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