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 =  8.3205 cm

Solution:

$$\begin{gathered} a:b=2:3 \\ b=3\mathrm{/}2\cdot a \\ c=10\text{ cm } \\ {c}^2=a^2+b^2 \\ {c}^2=a^2+(3\mathrm{/}2a{)}^2 \\ {c}^2=a^2+(3\mathrm{/}2{)}^2\cdot a^2 \\ a=\sqrt{{\displaystyle \frac{c^2}{1+(3\mathrm{/}2{)}^2}}}=\sqrt{{\displaystyle \frac{10^2}{1+(3\mathrm{/}2{)}^2}}}=5.547\text{ cm} \end{gathered}$$

$b=3\mathrm{/}2\cdot a=3\mathrm{/}2\cdot 5.547=8.3205\text{ cm}$

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