Right Triangle: Given hypotenuse and ratio of legs, find legs
We are given a Right triangle where the Hypotenuse = $20$ cm. The opposite side is $3$ times longer than the bottom side. Is it possible to calculate the length of the opposite side? (Tried substitution) $$a^2 + b^2 = 400$$ $$a = 3b$$ $$(3b)^2 + b^2 = 400$$ $b = 10$ = not correct
$\endgroup$ 43 Answers
$\begingroup$Suppose the length of the hypotenuse is $c$ and the other two sides have lengths $a$ and $b$.
We know that $c^2 = a^2+b^2$.
If $\dfrac{a}{b} = r$, then $a = br$ so that $c^2 = a^2+b^2 = (br)^2+b^2 = b^2(r^2+1) $.
Therefore, if you know $c$ and $r$, $b^2 =\dfrac{c^2}{r^2+1} $ and $a^2 = b^2r^2 =\dfrac{c^2r^2}{r^2+1} $.
In your case, $c=20$ and $r = 3$, so $b^2 =\dfrac{20^2}{10} =40 $ and $a^2 = b^2r^2 =40\cdot 9 =360 $.
$\endgroup$ $\begingroup$If $a=3b$ then,
$$(3b)^2+b^2=400\to 10b^2=400 \to b=2\sqrt{10}\to a=6\sqrt{10}$$
$\endgroup$ $\begingroup$Notice here. How is it?
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