Question

1. in a right triangle, one acute angle is 45°, one right side is $5\sqrt{3}$, the hypotenuse is $x$, and the other right side is $y$.

Explanation:

Step1: Identify triangle type

This is a 45-45-90 right triangle, so the two legs are equal.

Step2: Solve for $y$

Since legs are equal, $y = 5\sqrt{3}$

Step3: Solve for hypotenuse $x$

Use Pythagorean theorem: $x = \sqrt{(5\sqrt{3})^2 + (5\sqrt{3})^2}$
Simplify:
$x = \sqrt{75 + 75} = \sqrt{150} = 5\sqrt{6}$

Answer:

$x = 5\sqrt{6}$, $y = 5\sqrt{3}$