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. Given one leg is \(x\), the other leg is also \(x\).
Let the hypotenuse be \(c\).

Step2: Apply Pythagorean theorem

For a right - triangle \(a^{2}+b^{2}=c^{2}\). Here \(a = x\), \(b = x\). So \(x^{2}+x^{2}=c^{2}\), which simplifies to \(2x^{2}=c^{2}\).

Step3: Solve for hypotenuse

Taking the square root of both sides, \(c=\sqrt{2x^{2}}=\sqrt{2}x\).

Answer:

The other leg is \(x\) and the hypotenuse is \(\sqrt{2}x\)