Question

find the length of side a in the picture, below.
a =
enter your answer in exact form.

Explanation:

Step1: Identify Triangle Type

The triangle is right - angled with one angle \(45^{\circ}\), so it's a 45 - 45 - 90 triangle (isosceles right triangle), meaning the two legs are equal, and hypotenuse \(h = l\sqrt{2}\) (where \(l\) is leg length). But here, we can also use trigonometry. Let's use sine: \(\sin(45^{\circ})=\frac{a}{1}\) (since hypotenuse is 1, opposite side to \(45^{\circ}\) is \(a\)).

Step2: Recall Sine of \(45^{\circ}\)

We know that \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\). So from \(\sin(45^{\circ})=\frac{a}{1}\), we get \(a = \sin(45^{\circ})=\frac{\sqrt{2}}{2}\). Also, since it's a 45 - 45 - 90 triangle, the legs are equal. Let the legs be \(a\) and \(b\), hypotenuse \(c = 1\). By Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), and since \(a = b\), \(2a^{2}=1\), \(a^{2}=\frac{1}{2}\), \(a=\frac{\sqrt{2}}{2}\) (taking positive root as length).

Answer:

\(\frac{\sqrt{2}}{2}\)