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 the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the two legs are of equal length.
If one leg has length $x$, the other leg also has length $x$.

Step2: Find the hypotenuse length

The ratio of the sides of a 45 - 45 - 90 triangle is $1:1:\sqrt{2}$. If the length of each leg is $x$, then using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$ (where $a = b=x$), we have $x^{2}+x^{2}=c^{2}$, $2x^{2}=c^{2}$, and $c = x\sqrt{2}$.

Answer:

The length of the other leg is $x$, and the length of the hypotenuse is $x\sqrt{2}$