Question

a triangle with a right angle, one side labeled 4, another side labeled y, the base labeled x, and an angle of 45°. options: a) x = 4√2, y = 2√2; b) x = (4√6)/3, y = 2√2; c) x = 4√2, y = 4; d) x = (4√6)/3, y = 4. choose the correct option among a, b, c, d.

Explanation:

Step1: Identify Triangle Type

The triangle is right - angled (one right angle) and has a \(45^{\circ}\) angle. So, the third angle is \(180^{\circ}-90^{\circ}-45^{\circ}=45^{\circ}\). Thus, it is an isosceles right - triangle, meaning the two legs are equal. One leg is given as \(4\), so the other leg \(y = 4\).

Step2: Calculate Hypotenuse \(x\)

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the Pythagorean theorem is \(c=\sqrt{a^{2}+b^{2}}\). In an isosceles right - triangle with legs of length \(a = b=4\), the hypotenuse \(x=\sqrt{4^{2}+4^{2}}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).

Answer:

C. \(x = 4\sqrt{2},y = 4\)