How to find the length of the sides of an isosceles right triangle given its hypotenuse ? (Without trigonometry)

Question

I was doing my math homework when this question came.

There's an isosceles triangle with its base length which equals 13, and the other sides' length n. The question is, "if this triangle was a right triangle, what's the value of n ?".

PS. I'm a 9th grader but we still haven't seen trigonometry yet.

Answer 1

First note that every isosceles right triangle is similar. They all have the same shape, they're just scaled differently.

Now consider the isosceles right triangle with the two equal sides being $1$ unit. These have to be the mutually perpendicular sides (technically called the catheti) because the hypotenuse is strictly longer than any other side. Apply Pythagoras' Theorem. The hypotenuse $h$ is given by $h^2 = 1^2 + 1^2$, giving $h = \sqrt 2$.

You now know the ratios of the sides of every isosceles right triangle, i.e. $1:1:\sqrt 2$.

To apply it, if you're given that the hypotenuse is $13$, then the other two sides are each $\frac{13}{\sqrt 2} $. This is often rearranged to $\frac{13\sqrt 2}{2} $ to avoid an irrational number in the denominator.

Answer 2

By the Pythagorean theorem, $$n^2+n^2=13^2$$ $$2n^2=169$$ $$n^2=\frac{169}2$$ $$n=\frac{13}{\sqrt2}=9.192\dots$$

Answer 3

Hint: try to use Pythagoras' Theorem (https://en.wikipedia.org/wiki/Pythagorean_theorem).