Question

find the lengths of a, b, c, and d shown in the figure below if a + b = 27. round your answers to the nearest tenth. note that the figure is not drawn to scale.

Explanation:

Step1: Use cosine to find a

In the right - triangle with hypotenuse 7 and angle 57°, $\cos(57^{\circ})=\frac{a}{7}$, so $a = 7\cos(57^{\circ})$.
$a=7\times0.5446\approx3.8$

Step2: Find b

Since $a + b=27$, then $b = 27 - a$.
$b=27 - 3.8=23.2$

Step3: Use sine to find c

$\sin(57^{\circ})=\frac{c}{7}$, so $c = 7\sin(57^{\circ})$.
$c=7\times0.8387\approx5.9$

Step4: Use Pythagorean theorem to find d

In the right - triangle with legs c and b, by the Pythagorean theorem $d=\sqrt{c^{2}+b^{2}}$.
$d=\sqrt{(5.9)^{2}+(23.2)^{2}}=\sqrt{34.81 + 538.24}=\sqrt{573.05}\approx23.9$

Answer:

$a\approx3.8$
$b\approx23.2$
$c\approx5.9$
$d\approx23.9$