Question

1. using the patterns seen in problems 1 -6, to fill in the blank spaces of this triangle with side length x.

Explanation:

Step1: Identify triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the two legs are of equal length.

Step2: Determine unknown side length

Since one leg has length $x$, the other leg also has length $x$.

Step3: Find hypotenuse length

Use the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = x$, $b = x$. Then $x^{2}+x^{2}=c^{2}$, $2x^{2}=c^{2}$, and $c=\sqrt{2x^{2}}=\sqrt{2}x$.

Answer:

The other leg has length $x$ and the hypotenuse has length $\sqrt{2}x$.