# RT and ratio

A right triangle whose legs are in a ratio 6:12 has hypotenuse 68 m long. How long are its legs?

Result

a =  30.411 m
b =  60.821 m

#### Solution:

$c = 68\ m \ \\ a^2+b^2 = c^2 \ \\ a:b = 6:12 \ \\ a = 6/12 \cdot b \ \\ (6/12)^2b^2 + b^2 = c^2 \ \\ b^2(1+(6/12)^2) = c^2 \ \\ b = \dfrac{c}{ \sqrt{ 1+ (6/12)^2}} \ \\ b = \dfrac{ 68 }{ \sqrt{ 1+ (6/12)^2}} = 60.821 \ m \ \\ a = (6/12) \cdot b = 30.411 \ \text{m}$
$b=68 / \sqrt{ 1+6 \cdot \ 6/12/12 }=60.821 \ \text{m}$

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