Question

round your answers to the nearest tenth.
(a) (d=)
(b) (b=)

Explanation:

Step1: Solve for \(d\) in the 45 - 45 - 90 triangle

In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\). If the hypotenuse \(c = 2\), and the legs are of equal length \(d\), then \(c=\sqrt{2}d\). So, \(d=\frac{c}{\sqrt{2}}\). Substituting \(c = 2\), we get \(d=\frac{2}{\sqrt{2}}=\sqrt{2}\approx1.4\).

Step2: Solve for \(b\) in the 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, if the side opposite the 30 - degree angle is \(a\), the side opposite the 60 - degree angle is \(\sqrt{3}a\), and the hypotenuse is \(2a\). Here, the side opposite the 60 - degree angle is 6, and the side opposite the 30 - degree angle is \(a\). Also, \(\tan60^{\circ}=\frac{b}{6}\), and since \(\tan60^{\circ}=\sqrt{3}\), we have \(b = 6\tan60^{\circ}=6\sqrt{3}\approx10.4\).

Answer:

(a) \(d\approx1.4\)
(b) \(b\approx10.4\)