# Get size of cathetus/i from size of hypothenuse and also proportion in between catheti?

Just how can I compute the size of the cathetus in a triangular if I recognize the size of the hypotenuse and also the proportion in between both catheti?

As an example :

The hypotenuse is $5$cm long, and also the proportion in between the catheti is $4:3$ - how much time are the catheti or either cathetus?

You can call one cathetus 4 *x *, the various other 3 *x *, and also use Pythagora's theory : (4 *x *) ^{2 }+(3 *x *) ^{2 } = 5 ^{2 }.

You will certainly get 25 *x * ^{2 } = 25, which generates *x * = 1. So one cathetus is 4cm, and also the various other 3cm.

(Remember that 3,4,5 is a Pythagorean three-way)

Let the proportion be 1 :r and also the hypotenuse be h. After that the sides are after that x and also 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 after that compute rx, the opposite side.

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