Question

Use the markings in the image to answer the following questions (drawing not to scale):

Answer 1

a. We can calculate all the sides using the given information.

b. 
- Length of $a$:
The first thing we are going to do is find the value of theta; to do it we are going to use arcosine: 
$cos \alpha = \frac{5}{13}$
$\alpha =arcos( \frac{5}{13} )$
$\alpha =67.38$

Next, since we know the the length of adjacent side of theta is 12, we can use the trig function cosine to find the length of $a$:
$cos( \alpha )= \frac{adjacent.side}{hypotenuse}$
$cos(67.38)= \frac{12}{a}$
$a= \frac{12}{cos(67.38)}$
$a=31.20$

- Length of $b$:
Since we already find the length of $a$, we just need to use the Pythagorean theorem to find the length of $b$:
$b^2=a^2-12^2$
$b^2=31.20^2-12^2$
$b= \sqrt{31.20^2-12^2}$
$b=28.8$

- Length of $c$
We already know the measure of angle theta, so we are going to use the trig function tangent to find the length of $b$:
$tan( \alpha )= \frac{opposite.side}{adjacent.side}$
$tan(67.38)= \frac{12}{c}$
$c= \frac{12}{tan(67.38)}$
$c=5$

- Length of $d$:
Since we already find the length of $c$, we are going to use the Pythagorean theorem to find the length of $d$:
$d^2=c^2+12^2$
$d^2=5^2+12^2$
$d= \sqrt{5^2+12^2}$
$d=13$

c. We can conclude that we have all the information we need to find the measures of all the sides of our triangles.