Question

find the lengths of a, b, c, and d shown in the figure below if a + b = 21. round your answers to the nearest tenth. note that the figure is not drawn to scale.

Explanation:

Step1: Find $a$ using cosine function

In the right - triangle with hypotenuse 5 and angle $55^{\circ}$, $\cos(55^{\circ})=\frac{a}{5}$, so $a = 5\cos(55^{\circ})\approx5\times0.574 = 2.87$.

Step2: Find $b$

Since $a + b=21$, then $b=21 - a$. Substituting $a\approx2.87$, we get $b = 21-2.87 = 18.13$.

Step3: Find $c$ using sine function

In the right - triangle with hypotenuse 5 and angle $55^{\circ}$, $\sin(55^{\circ})=\frac{c}{5}$, so $c = 5\sin(55^{\circ})\approx5\times0.819 = 4.095\approx4.1$.

Step4: Find $d$ using Pythagorean theorem

In the large right - triangle with legs $b = 18.13$ and $c\approx4.1$, by the Pythagorean theorem $d=\sqrt{b^{2}+c^{2}}=\sqrt{(18.13)^{2}+(4.1)^{2}}=\sqrt{328.6969 + 16.81}=\sqrt{345.5069}\approx18.6$.

Answer:

$a\approx2.9$
$b\approx18.1$
$c\approx4.1$
$d\approx18.6$