# Right triangle - ratio

The lengths of the legs of the right triangle ABC are in ratio b = 2: 3. The hypotenuse is 10 cm long. Calculate the lengths of the legs of that triangle.

Correct result:

a =  5.547 cm
b =  8.321 cm

#### Solution:

$a:b=2:3 \ \\ b=3/2 \cdot \ a \ \\ c=10 \ \text{cm} \ \\ \ \\ \ \\ c^2=a^2+b^2 \ \\ c^2=a^2+(3/2a)^2 \ \\ c^2=a^2+(3/2)^2 \cdot \ a^2 \ \\ a=\sqrt{ \dfrac{ c^2 }{ 1+(3/2)^2 } }=\sqrt{ \dfrac{ 10^2 }{ 1+(3/2)^2 } }=5.547 \ \text{cm}$
$b=3/2 \cdot \ a=3/2 \cdot \ 5.547=8.321 \ \text{cm}$

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