Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).

• A. $x=81$
• B. $x=9\sqrt2$
• C. $x=2\sqrt9$
• D. $x=\sqrt{162}$

Answer: (B)

Verified step by step guidance

1. Identify the type of triangle: The given triangle is a right triangle with one angle measuring 45 degrees. This indicates that it is a 45-45-90 triangle, which is an isosceles right triangle.
2. Recall the properties of a 45-45-90 triangle: In a 45-45-90 triangle, the legs are congruent, and the hypotenuse is \( \sqrt{2} \) times the length of each leg.
3. Given that both legs of the triangle are 9 units, use the property of 45-45-90 triangles to find the hypotenuse. The formula for the hypotenuse \( x \) is \( x = 9 \times \sqrt{2} \).
4. Simplify the expression for the hypotenuse: \( x = 9 \sqrt{2} \). This is the length of the hypotenuse in its simplest form.
5. Verify the solution: Check that the calculated hypotenuse \( x = 9 \sqrt{2} \) is consistent with the properties of a 45-45-90 triangle, ensuring that the solution is correct.