Question

solve for a.
(image of a right triangle with one leg labeled (2sqrt{2}), one angle (45^circ), hypotenuse (a), and the other leg (b))

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (one angle is \(90^{\circ}\), another is \(45^{\circ}\), so the third angle is also \(45^{\circ}\)). In a right - isosceles triangle, the legs are equal, and the hypotenuse \(a\) can be found using the Pythagorean theorem or trigonometric ratios. Let's use trigonometric ratios. We know that \(\sin(45^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\), the opposite side to the \(45^{\circ}\) angle is \(2\sqrt{2}\), and the hypotenuse is \(a\). Also, \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\).

Step2: Apply Sine Formula

We have \(\sin(45^{\circ})=\frac{2\sqrt{2}}{a}\). Substituting \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\) into the equation, we get \(\frac{\sqrt{2}}{2}=\frac{2\sqrt{2}}{a}\). Cross - multiply: \(a\times\sqrt{2}=2\times2\sqrt{2}\).

Step3: Solve for \(a\)

Simplify the right - hand side: \(a\times\sqrt{2} = 4\sqrt{2}\). Divide both sides by \(\sqrt{2}\), we get \(a = 4\).

Answer:

\(a = 4\)