Question

determine the length of the legs for the given 45 - 45 - 90 triangle:

Explanation:

Step1: Recall the properties of a 45 - 45 - 90 triangle

In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\). Let the length of each leg be \(x\). The hypotenuse \(c\) is related to the leg \(a\) (or \(b\)) by the formula \(c = a\sqrt{2}\). Here, the hypotenuse \(AB = 32\sqrt{3}\). Wait, no, correction: assume the hypotenuse is \(h\) and legs are \(l\). The formula for a 45 - 45 - 90 triangle is \(h=\sqrt{2}l\). But wait, maybe there was a mis - read. Wait, no, actually, if we assume the given side is the hypotenuse. Wait, no, wait, in a right - isosceles triangle (45 - 45 - 90), if the hypotenuse is \(h\) and legs are \(l\), then \(h = l\sqrt{2}\). But if we assume that the given side is the hypotenuse. Wait, no, wait, let's re - check.

Wait, actually, in a 45 - 45 - 90 triangle, if the legs are of length \(a\), then the hypotenuse \(c=a\sqrt{2}\). But if we assume that the given side is the hypotenuse. Wait, no, wait, hold on. Wait, the problem is to find the legs. Let's use the Pythagorean theorem. Let the legs be \(x\) (since in a 45 - 45 - 90 triangle, the two legs are equal). By Pythagorean theorem \(x^{2}+x^{2}=h^{2}\), \(2x^{2}=h^{2}\), \(x = \frac{h}{\sqrt{2}}\). But if \(h = 32\sqrt{3}\), no, wait, no, wait, looking back at the options, maybe there was a mis - take in reading the side. Wait, no, wait, hold on. Wait, the problem is likely that the given side is the hypotenuse. Wait, no, wait, in 45 - 45 - 90 triangle, legs \(l\), hypotenuse \(h = l\sqrt{2}\). If we solve for \(l\), \(l=\frac{h}{\sqrt{2}}\). But if \(h = 32\sqrt{3}\), no, that's not matching the options. Wait, no, wait, looking at the options \(16\sqrt{6}\). Let's check: if \(l = 16\sqrt{6}\), then \(h=l\sqrt{2}=16\sqrt{6}\times\sqrt{2}=16\sqrt{12}=32\sqrt{3}\)

Answer:

\(16\sqrt{6}\)