for the right triangles below, find the exact values of the side lengths h and b. if necessary, write your…

Explanation:

Step1: Find h using 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the side opposite the 30° angle is $x$, the side opposite the 60° angle is $x\sqrt{3}$ and the hypotenuse is $2x$. Here the side opposite the 30° angle is $h$, and the side opposite the 60° angle is 5. So, $\tan60^{\circ}=\frac{5}{h}$, and since $\tan60^{\circ}=\sqrt{3}$, we have $\sqrt{3}=\frac{5}{h}$, then $h = \frac{5}{\sqrt{3}}=\frac{5\sqrt{3}}{3}$.

Step2: Find b using 45 - 45 - 90 triangle ratio

In a 45 - 45 - 90 triangle, the two legs are equal and if each leg is $a$, the hypotenuse is $a\sqrt{2}$. Here one leg is 2, and using the Pythagorean theorem $b^{2}=2^{2}+2^{2}$ (or simply knowing the ratio for 45 - 45 - 90 triangle), $b=\sqrt{2^{2}+2^{2}}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$.

Answer:

$h=\frac{5\sqrt{3}}{3}$ $b = 2\sqrt{2}$