Question

fill in the blank 1 point
if y=8, find x and z.
x = choose your answer... z = choose your answer...

Explanation:

Step1: Identify the triangle type

This is a right - isosceles triangle (one angle is \(90^{\circ}\), another is \(45^{\circ}\), so the third angle is also \(45^{\circ}\)). In a right - isosceles triangle, the two legs (the sides opposite the \(45^{\circ}\) angles) are equal, and the hypotenuse \(y\) is related to the leg \(x\) (or \(z\)) by the formula \(y = x\sqrt{2}\) (or \(y=z\sqrt{2}\)), and \(x = z\).

Step2: Solve for \(x\) (and \(z\) since \(x = z\))

We know that \(y = 8\) and \(y=x\sqrt{2}\). So we can solve for \(x\) by rearranging the formula: \(x=\frac{y}{\sqrt{2}}\). Rationalizing the denominator, we multiply the numerator and denominator by \(\sqrt{2}\), so \(x = \frac{y\sqrt{2}}{2}\). Substituting \(y = 8\), we get \(x=\frac{8\sqrt{2}}{2}=4\sqrt{2}\). Since \(x = z\) (because it's a right - isosceles triangle with two \(45^{\circ}\) angles, so the two legs are equal), \(z = 4\sqrt{2}\) as well. Wait, no, wait. Wait, in a right - isosceles triangle, if the hypotenuse is \(y\), and the legs are \(x\) and \(z\), and the angles are \(45^{\circ},45^{\circ},90^{\circ}\), then the ratio of the sides is \(x:z:y=1:1:\sqrt{2}\). So if \(y = 8\), then \(x=z=\frac{y}{\sqrt{2}}=\frac{8}{\sqrt{2}} = 4\sqrt{2}\)? Wait, no, that's a mistake. Wait, actually, if the legs are \(a\) and \(a\), and the hypotenuse is \(a\sqrt{2}\). So if the hypotenuse \(y=a\sqrt{2}=8\), then \(a=\frac{8}{\sqrt{2}} = 4\sqrt{2}\)? Wait, no, let's do it correctly. Let's consider the triangle: the right angle is between \(x\) and \(z\), so \(x\) and \(z\) are the legs, \(y\) is the hypotenuse. The angle opposite to \(x\) is \(45^{\circ}\), and the angle opposite to \(z\) is \(45^{\circ}\), so \(x = z\) (because in a triangle, sides opposite equal angles are equal). And by Pythagoras theorem, \(x^{2}+z^{2}=y^{2}\). Since \(x = z\), we have \(2x^{2}=y^{2}\). Substituting \(y = 8\), we get \(2x^{2}=64\), so \(x^{2}=32\), so \(x=\sqrt{32}=4\sqrt{2}\), and \(z = 4\sqrt{2}\). Wait, but another way: in a \(45 - 45 - 90\) triangle, the legs are equal, and the hypotenuse is leg \(\times\sqrt{2}\). So if the hypotenuse is \(y = 8\), then leg \(x=\frac{y}{\sqrt{2}}=\frac{8}{\sqrt{2}}=4\sqrt{2}\), and since \(x = z\), \(z = 4\sqrt{2}\). Wait, but maybe I mixed up the sides. Wait, no, the angle at the bottom is \(45^{\circ}\), so the side opposite to it is \(x\), and the side adjacent to it is \(z\). Wait, no, the right angle is between \(x\) and \(z\), so \(x\) is vertical leg, \(z\) is horizontal leg, \(y\) is hypotenuse. The angle at the bottom is \(45^{\circ}\), so \(\sin(45^{\circ})=\frac{x}{y}\) and \(\cos(45^{\circ})=\frac{z}{y}\). Since \(\sin(45^{\circ})=\cos(45^{\circ})=\frac{\sqrt{2}}{2}\), then \(x = y\times\sin(45^{\circ})=8\times\frac{\sqrt{2}}{2}=4\sqrt{2}\), and \(z = y\times\cos(45^{\circ})=8\times\frac{\sqrt{2}}{2}=4\sqrt{2}\).

Wait, but maybe the triangle is labeled differently. Wait, maybe the legs are \(x\) and \(z\), and the hypotenuse is \(y\), and the angles are \(45^{\circ}\) (at the bottom), \(90^{\circ}\) (at the left), so the angle at the top is \(45^{\circ}\). So it's a \(45 - 45 - 90\) triangle, so the two legs are equal, and hypotenuse is leg \(\times\sqrt{2}\). So if hypotenuse \(y = 8\), then each leg \(x=z=\frac{y}{\sqrt{2}} = 4\sqrt{2}\).

Answer:

\(x = 4\sqrt{2}\), \(z = 4\sqrt{2}\)