The given triangle is not a right angle triangle. We would solve for r, the unknown by applying the cosine rule which is expressed as
a^2 = b^2 + c^2 - 2bcCosA
By relating the given triangle with the cosine rule,
a = r = PQ
b = 9 = PR
c = 7.5 = QR
angle A = angle R = 89 degrees
Thus, we have
r^2 = 9^2 + 7.5^2 -2 * 9 * 7.5 * Cos89
r^2 = 81 + 56.25 - 135Cos89
r^2 = 137.25 - 135 * 0.01745
r^2 = 137.25 - 2.35575
r^2 = 134.89425
[tex]\begin{gathered} r\text{ = }\sqrt[]{134.89425} \\ r\text{ = 11.61} \end{gathered}[/tex]
To the nearest tenth,
r = 11.6
We would solve for angle P by applying the Sine rule which is expressed as
a/SinA = b/SinB
Bt appying it on the triangle, we have
pQ/SinR = QR/SinP
11.6/Sin89 = 7.5/SinP
By crossmultiplying, it becomes
11.6SinP = 7.5Sin89
11.6SinP = 7.4989
SinP = 7.4989/11.6
SinP = 0.6465
P = Sin^-1(0.6465)
P = 40.3 degrees
Recall, the sum of the angles in a triangle is 180 degrees. Thus, we have
P + Q + R = 180
40.3 + Q + 89 = 180
Q + 129.3 = 180
Q = 180 - 129.3
Q = 50.7 degrees
