Question

solve for x
(right triangle with one angle 45°, one leg is \\(\frac{3\sqrt{2}}{2}\\), hypotenuse x, other leg y)

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (one right angle, one \(45^{\circ}\) angle, so the third angle is also \(45^{\circ}\)). In a \(45 - 45 - 90\) triangle, the legs are equal, and the hypotenuse \(x\) is related to the leg \(l\) by the formula \(x = l\sqrt{2}\), and the leg \(l\) is related to the hypotenuse by \(l=\frac{x}{\sqrt{2}}\) or \(l = x\sin(45^{\circ})\) or \(l=x\cos(45^{\circ})\) (since \(\sin(45^{\circ})=\cos(45^{\circ})=\frac{\sqrt{2}}{2}\)). Here, one of the legs is \(\frac{3\sqrt{2}}{2}\).

Step2: Use Trigonometric Ratio or 45 - 45 - 90 Triangle Properties

We know that in a right - triangle, \(\cos(45^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\). The adjacent side to the \(45^{\circ}\) angle is \(\frac{3\sqrt{2}}{2}\), and the hypotenuse is \(x\). Since \(\cos(45^{\circ})=\frac{\sqrt{2}}{2}\), we have the equation \(\frac{\sqrt{2}}{2}=\frac{\frac{3\sqrt{2}}{2}}{x}\).

Cross - multiply: \(\frac{\sqrt{2}}{2}\times x=\frac{3\sqrt{2}}{2}\)

Multiply both sides by 2: \(\sqrt{2}x = 3\sqrt{2}\)

Divide both sides by \(\sqrt{2}\): \(x = 3\)

Alternatively, using the property of \(45 - 45 - 90\) triangle: If the leg length is \(l\), hypotenuse \(h=l\sqrt{2}\). Here, \(l = \frac{3\sqrt{2}}{2}\), so \(h=\frac{3\sqrt{2}}{2}\times\sqrt{2}=\frac{3\times2}{2}=3\)

Answer:

\(x = 3\)