Question

find the lengths of a, b, c, and d shown in the figure below if ac = 40. round your answers to the nearest tenth. note that the figure is not drawn to scale.

Explanation:

Step1: Use trigonometric relations

In the right - triangle with angle $27^{\circ}$, if we consider the side of length $13$ as the hypotenuse. Let's assume the vertical side of the right - triangle with angle $27^{\circ}$ is $a$. Then $\sin(27^{\circ})=\frac{a}{13}$, so $a = 13\sin(27^{\circ})\approx13\times0.454\approx5.9$.

Step2: Find $c$ using $ac = 40$

Since $ac = 40$ and $a\approx5.9$, then $c=\frac{40}{a}=\frac{40}{5.9}\approx6.8$.

Step3: Find $b$

In the right - triangle with angle $27^{\circ}$, $\cos(27^{\circ})=\frac{b}{13}$, so $b = 13\cos(27^{\circ})\approx13\times0.891\approx11.6$.

Step4: Find $d$

Using the Pythagorean theorem in the large right - triangle with sides $b$ and $(a + c)$. First, $a + c\approx5.9+6.8 = 12.7$. Then $d=\sqrt{b^{2}+(a + c)^{2}}=\sqrt{11.6^{2}+12.7^{2}}=\sqrt{134.56+161.29}=\sqrt{295.85}\approx17.2$.

Answer:

$a\approx5.9$
$b\approx11.6$
$c\approx6.8$
$d\approx17.2$