Question

find the value of each variable. if your answer is not an integer, write it in simplest radical form with the denominator rationalized.
$y=\square$ and $z=\square$
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal: $y = z$.

Step2: Apply Pythagorean theorem

For right triangle, $y^2 + z^2 = (18\sqrt{2})^2$. Substitute $z=y$:
$$2y^2 = (18\sqrt{2})^2$$

Step3: Calculate right-hand side

$$(18\sqrt{2})^2 = 18^2 \times (\sqrt{2})^2 = 324 \times 2 = 648$$

Step4: Solve for $y$

$$2y^2 = 648 \implies y^2 = 324 \implies y = \sqrt{324} = 18$$

Step5: Find $z$

Since $y=z$, $z=18$.

Answer:

$y = 18$ and $z = 18$