Question

find the length of side z in the picture, below.
triangle image with 45° angle, right angle, hypotenuse 1, side z
z =
enter your answer in exact form.
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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 the hypotenuse \(c\) and leg \(a\) have the relation \(c=a\sqrt{2}\), or leg \(a = \frac{c}{\sqrt{2}}\) (rationalizing the denominator, \(a=\frac{c\sqrt{2}}{2}\)). Here, the hypotenuse is \(1\), and \(z\) is a leg.

Step2: Apply Trigonometry (or Special Triangle Ratio)

Using cosine: \(\cos(45^{\circ})=\frac{z}{1}\). Since \(\cos(45^{\circ})=\frac{\sqrt{2}}{2}\), we have \(z = \cos(45^{\circ})\times1=\frac{\sqrt{2}}{2}\). Or using the \(45 - 45 - 90\) triangle ratio: leg \(=\frac{\text{hypotenuse}}{\sqrt{2}}\), so \(z=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\) (after rationalizing the denominator).

Answer:

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