Hello!
To find the value of a r, first we can find the angle R, and then use the law of sines.
The law of sines states: a / sin A = b / sin B = c / sin C.
In this case, it is also: e / sin E = p / sin P = r / sin R.
1. Find the measure of angle R.
Since triangles are equal to 180 degrees, we add the angle of P and E together, and subtract that from 180, and solve.
45 + 63 + R = 180
108 + R = 180 (subtract 108 from both sides)
R = 72
With the measure of angle R, we can find the length of side r.
2. Use the Law of Sines.
22 / sin 45 = r / sin 72 (multiply both sides by sin 72)
r = 22 sin 72 / sin 45 (simplify)
r = 29.58993...
This can be rounded to 29.59.
Therefore, the length of side r is 29.59.