Question

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

Explanation:

Step1: Identify triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides to the hypotenuse is $1:1:\sqrt{2}$. Let the length of each of the legs be $a$. Then, by the Pythagorean theorem $a^{2}+a^{2}=x^{2}$, which simplifies to $2a^{2}=x^{2}$, and $a = \frac{x}{\sqrt{2}}=\frac{\sqrt{2}x}{2}$.

Answer:

Each of the two legs of the triangle has length $\frac{\sqrt{2}x}{2}$