Question

xi sketches a right triangle with legs of lengths x and y and a hypotenuse of length z. she writes an equation that relates x, y, and z. then, she writes an expression for the length of the hypotenuse. which expression does she write?
options:

• $\sqrt{x^2 + y^2}$
• $x^2 - y^2$
• $\sqrt{x^2 + z^2}$
• $\sqrt{x + y}$

Explanation:

Step1: Recall Pythagorean Theorem

For a right triangle with legs \(x\), \(y\) and hypotenuse \(z\), the Pythagorean theorem states \(x^{2}+y^{2}=z^{2}\).

Step2: Solve for \(z\)

To find the expression for \(z\) (the hypotenuse), we take the square root of both sides of the equation \(x^{2}+y^{2}=z^{2}\). So \(z = \sqrt{x^{2}+y^{2}}\) (since length is non - negative, we take the positive square root).

Answer:

\(\boldsymbol{\sqrt{x^{2}+y^{2}}}\) (the first option)