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How can I calculate the length of the cathetus in a triangle if I know the length of the hypotenuse and the ratio between the two catheti?

For example:

The hypotenuse is $5$cm long, and the ratio between the catheti is $4:3$ - how long are the catheti or either cathetus?

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    Most people would use the word "leg" rather than "cathetus", which I'd never seen before. Just letting you know, because it might help you in the future.2010-08-05
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    Thanks! Used Wikipedia to translate the word.2010-08-05

2 Answers 2

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Let the ratio be 1:r and the hypotenuse be h. Then the sides are then x and rx for some x. By the Pythagorean Theorem we get $x^2+r^2*x^2=h^2$. So $x=\sqrt{{h^2}/(1+r^2)}$. We can then calculate rx, the other side.

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You can call one cathetus 4*x*, the other 3*x*, and apply Pythagora's theorem: (4*x*)2+(3*x*)2=52.

You will obtain 25*x*2=25, which yields x=1. So one cathetus is 4cm, and the other 3cm.

(Remember that 3,4,5 is a Pythagorean triple)

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    How can I calculate it if the ratio is not constant?2010-08-05
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    Pythagorean theorem..2010-08-05