Question

solve for x. image of a right triangle with one angle 45°, one leg labeled 3√2, and hypotenuse labeled x

Explanation:

Step1: Identify Triangle Type

The triangle is a right - triangle with one angle \(45^{\circ}\). So, it's a \(45 - 45 - 90\) triangle (isosceles right - triangle), where the legs are equal, and the hypotenuse \(h\) is related to the leg \(l\) by \(h = l\sqrt{2}\). Here, the leg length is \(3\sqrt{2}\), and \(x\) is the hypotenuse.

Step2: Apply \(45 - 45 - 90\) Formula

For a \(45 - 45 - 90\) triangle, if the leg length is \(l\) and hypotenuse is \(h\), then \(h=l\sqrt{2}\). But we can also use the cosine function. In a right - triangle, \(\cos(45^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\). The adjacent side to the \(45^{\circ}\) angle is \(3\sqrt{2}\), and the hypotenuse is \(x\). We know that \(\cos(45^{\circ})=\frac{\sqrt{2}}{2}\). So, \(\frac{\sqrt{2}}{2}=\frac{3\sqrt{2}}{x}\). Cross - multiplying gives \(x\times\sqrt{2}=2\times3\sqrt{2}\). Dividing both sides by \(\sqrt{2}\), we get \(x = 6\).
Or using the \(45 - 45 - 90\) triangle ratio: leg : leg : hypotenuse \(=1:1:\sqrt{2}\). Let the leg be \(a = 3\sqrt{2}\), then hypotenuse \(x=a\sqrt{2}\). Substituting \(a = 3\sqrt{2}\), we have \(x = 3\sqrt{2}\times\sqrt{2}=3\times2 = 6\).

Answer:

\(x = 6\)