Answer:
m∠R = 50 degrees
PR ≈ 5.74 units
PQ ≈ 8.53 units
Step-by-step explanation:
Let's first find the measure of the third angle, which should be easier to find. Remember that the total degree measure of a triangle is 180, so all three angles should add up to 180. Here, we already have <RPQ = 99 and <PQR = 31. So, <PRQ = 180 - 99 - 31 = 50 degrees.
We need to use the law of sines to figure out the lengths of the other two sides. The law of sines states that for a triangle with angles A, B, and C and side lengths a, b, and c, respectively: [tex]\frac{a}{sinA} =\frac{b}{sinB} =\frac{c}{sinC}[/tex]
Here, we can say that a = PR, b = QR, c = PQ, A = <PQR, B = <RPQ, and C = <PRQ. Then, we have:
[tex]\frac{a}{sinA} =\frac{b}{sinB} =\frac{c}{sinC}[/tex]
[tex]\frac{PR}{sin(PQR)} =\frac{QR}{sin(RPQ)} =\frac{PQ}{sin(PRQ)}[/tex]
We know that QR = 11, <RPQ = 99, <PQR = 31, and <PRQ = 50, so we can try to find PR:
[tex]\frac{PR}{sin(PQR)} =\frac{QR}{sin(RPQ)}[/tex]
[tex]\frac{PR}{sin(31)} =\frac{11}{sin(99)}[/tex]
[tex]PR =\frac{11}{sin(99)}*sin(31)[/tex] ≈ 5.74 units
Now, let's find PQ:
[tex]\frac{QR}{sin(RPQ)} =\frac{PQ}{sin(PRQ)}[/tex]
[tex]\frac{11}{sin(99)} =\frac{PQ}{sin(50)}[/tex]
[tex]\frac{11}{sin(99)}*sin(50) =PQ[/tex] ≈ 8.53 units
Hope this helps!
