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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

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    Did you try anything else? – 2017-02-01
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    $(3b)^2 = 9b^2$ – 2017-02-01
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    Why would $(3b)^2 + b^2 = 400$ make you think that $b = 10$? That *will* give you the correct answer but it isn't 10. Hint: it's not rational. – 2017-02-01
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    (3b)^2 = 9b^2.. – 2017-02-01

3 Answers 3

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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 $.

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If $a=3b$ then,

$$(3b)^2+b^2=400\to 10b^2=400 \to b=2\sqrt{10}\to a=6\sqrt{10}$$

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Notice here. How is it?

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