Questions: For the right triangles below, find the exact values of the side lengths w and z. The figures are not drawn to scale. (a) w= (b) z=

Solution Steps

We are given a 30-60-90 right triangle with hypotenuse 1. Let the side opposite the 30° angle be \(x\). Let the side opposite the 60° angle be \(y\). The ratio of sides in a 30-60-90 triangle is \(x : x\sqrt{3} : 2x\). In our case, the hypotenuse is 1, which corresponds to \(2x\). So, \(2x = 1\), which implies \(x = \frac{1}{2}\). The side opposite the 30° angle is \(w\), so \(w = x = \frac{1}{2}\). The side opposite the 60° angle is \(y = x\sqrt{3} = \frac{1}{2}\sqrt{3} = \frac{\sqrt{3}}{2}\).

We are given a 45-45-90 right triangle with hypotenuse 1. Let the two legs of the triangle be \(a\). The ratio of sides in a 45-45-90 triangle is \(a : a : a\sqrt{2}\). In our case, the hypotenuse is 1, which corresponds to \(a\sqrt{2}\). So, \(a\sqrt{2} = 1\), which implies \(a = \frac{1}{\sqrt{2}}\). Multiplying the numerator and denominator by \(\sqrt{2}\), we get \(a = \frac{\sqrt{2}}{2}\). The side \(z\) is one of the legs, so \(z = a = \frac{\sqrt{2}}{2}\).

Final Answer