Question

find k.
8√3 yd
45°
k
45°
write your answer in simplest radical form.
yards

Explanation:

Step1: Identify triangle type

The triangle is a right - isosceles triangle (two angles are \(45^{\circ}\), so the two legs are equal, and hypotenuse \(k\) can be found using the relationship in a \(45 - 45-90\) triangle. In a \(45 - 45-90\) triangle, if the length of each leg is \(l\), the hypotenuse \(h\) is given by \(h = l\sqrt{2}\). Here, one of the legs is \(8\sqrt{3}\) yd.

Step2: Apply the formula for hypotenuse

We know that for a \(45 - 45-90\) triangle, hypotenuse \(k=\text{leg}\times\sqrt{2}\). The length of the leg is \(8\sqrt{3}\) yd. So we substitute the value of the leg into the formula:
\(k = 8\sqrt{3}\times\sqrt{2}\)
Using the property of radicals \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), we get \(k = 8\sqrt{3\times2}=8\sqrt{6}\)

Answer:

\(8\sqrt{6}\)