Question

Triangle P Q R is shown. Angle R P Q is 99 degrees and angle P Q R is 31 degrees. The length of Q R is 11. Determine the measures of all unknown angles and side lengths of ΔPQR. Round side lengths to the nearest hundredth. m∠R = ° PR ≈ PQ ≈

Answer 1

Answer:R=50 PR=5.74 PQ=8.53

Explanation:

On edge:)

Answer 2

m∠R = 50 degrees

PR ≈ 5.74 units

PQ ≈ 8.53 units

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:
`(a)/(sinA) =(b)/(sinB) =(c)/(sinC)`

Here, we can say that a = PR, b = QR, c = PQ, A = <PQR, B = <RPQ, and C = <PRQ. Then, we have:

`(a)/(sinA) =(b)/(sinB) =(c)/(sinC)`

`(PR)/(sin(PQR)) =(QR)/(sin(RPQ)) =(PQ)/(sin(PRQ))`

We know that QR = 11, <RPQ = 99, <PQR = 31, and <PRQ = 50, so we can try to find PR:

`(PR)/(sin(PQR)) =(QR)/(sin(RPQ))`

`(PR)/(sin(31)) =(11)/(sin(99))`

`PR =(11)/(sin(99))*sin(31)` ≈ 5.74 units

Now, let's find PQ:

`(QR)/(sin(RPQ)) =(PQ)/(sin(PRQ))`

`(11)/(sin(99)) =(PQ)/(sin(50))`

`(11)/(sin(99))*sin(50) =PQ` ≈ 8.53 units

Hope this helps!