Question

19)
45°
6
90°
y
x

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the legs are equal (\(x = y\)), and the hypotenuse \(c\) is related to the leg \(a\) by the formula \(c=a\sqrt{2}\). Here \(c = 6\).

Step2: Solve for \(x\) and \(y\)

From \(c=a\sqrt{2}\), we can solve for \(a\) (where \(a=x=y\)). Rearranging the formula gives \(a=\frac{c}{\sqrt{2}}\). Substitute \(c = 6\) into the formula: \(a=\frac{6}{\sqrt{2}}\). Rationalize the denominator: \(\frac{6}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=3\sqrt{2}\).

Answer:

\(x = 3\sqrt{2}\), \(y = 3\sqrt{2}\)