Question

the hypotenuse of a 45°-45°-90° triangle measures 128 cm. what is the length of one leg of the triangle? options: 64 cm, 128 cm, 128√2 cm, 64√2 cm

Explanation:

Step1: Recall 45-45-90 triangle ratios

In a \(45^\circ - 45^\circ - 90^\circ\) triangle, the legs are equal, and the hypotenuse \(c\) is related to a leg \(a\) by \(c = a\sqrt{2}\). So, \(a=\frac{c}{\sqrt{2}}\).

Step2: Substitute hypotenuse value

Given \(c = 128\) cm, substitute into the formula: \(a=\frac{128}{\sqrt{2}}\). Rationalize the denominator: \(a=\frac{128\sqrt{2}}{2}=64\sqrt{2}\) cm? Wait, no, wait—wait, no, let's check again. Wait, no, in 45-45-90, legs are \(a\), hypotenuse is \(a\sqrt{2}\). So if hypotenuse is \(128\), then \(a\sqrt{2}=128\), so \(a = \frac{128}{\sqrt{2}}=\frac{128\sqrt{2}}{2}=64\sqrt{2}\)? Wait, but wait, the options: first option is \(64\sqrt{2}\) cm, fourth is 64 cm. Wait, maybe I made a mistake. Wait, no—wait, 45-45-90 triangle: legs are equal, hypotenuse is leg * \(\sqrt{2}\). So if leg is \(x\), hypotenuse is \(x\sqrt{2}\). So \(x\sqrt{2}=128\), so \(x=\frac{128}{\sqrt{2}}=\frac{128\sqrt{2}}{2}=64\sqrt{2}\). Wait, but let's check the options. The first option is \(64\sqrt{2}\) cm. Wait, but maybe the problem is that the triangle has two equal legs (marked with ticks), so it's isoceles right triangle. So hypotenuse \(h = leg \times \sqrt{2}\), so leg \(= \frac{h}{\sqrt{2}}=\frac{128}{\sqrt{2}} = 64\sqrt{2}\) cm. So the first option is \(64\sqrt{2}\) cm. Wait, but let me re - calculate: \(\frac{128}{\sqrt{2}}=\frac{128\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{128\sqrt{2}}{2}=64\sqrt{2}\). Yes. So the length of one leg is \(64\sqrt{2}\) cm.

Answer:

\(64\sqrt{2}\) cm (corresponding to the option with text "64√2 cm")