Question

a right triangle with one angle 45°, one leg is $3\sqrt{2}$, hypotenuse is $x$, the other leg is $y$. options: a) $x = 6$, $y = 3\sqrt{6}$; b) $x = 3\sqrt{6}$, $y = 6$; c) $x = 6$, $y = 3\sqrt{2}$; d) $x = 3\sqrt{2}$, $y = 3\sqrt{6}$

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (one angle is \(45^{\circ}\), right - angled), so the two legs are equal, and the hypotenuse \(x\) is related to the leg by \(x = \text{leg}\times\sqrt{2}\). The leg length is \(3\sqrt{2}\).

Step2: Calculate \(y\)

Since it's a \(45 - 45 - 90\) triangle, the two legs are equal. So \(y=3\sqrt{2}\)? Wait, no, wait. Wait, the angle given is \(45^{\circ}\), the right angle, so the other non - right angle is also \(45^{\circ}\) (since the sum of angles in a triangle is \(180^{\circ}\), \(180 - 90 - 45=45\)). So the two legs are equal. Wait, the leg given is \(3\sqrt{2}\), so \(y = 3\sqrt{2}\)? Wait, no, wait, let's recast. In a \(45 - 45 - 90\) triangle, the ratio of sides is \(1:1:\sqrt{2}\) (leg:leg:hypotenuse). Let the leg be \(a\), then hypotenuse \(c=a\sqrt{2}\). Here, one leg is \(3\sqrt{2}\), so the other leg \(y = 3\sqrt{2}\), and the hypotenuse \(x=(3\sqrt{2})\times\sqrt{2}\). Calculate \(x\): \((3\sqrt{2})\times\sqrt{2}=3\times2 = 6\). So \(x = 6\) and \(y = 3\sqrt{2}\), which is option C.

Answer:

C. \(x = 6\), \(y=3\sqrt{2}\)