In triangle PQR, mZP = 53°, PQ = 7.4, and PR = 9.6. What is m≤R to the nearest degree?
61°
49°
42°
35°
To find the measure of angle R, we can use the Law of Cosines.
The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the two sides and the cosine of the included angle.
Let's denote angle R as angle QPR. Using the Law of Cosines, we can write:
PR² = PQ² + QR² - 2(PQ)(QR)cos(QPR)
Substituting the given values, we have:
9.6² = 7.4² + QR² - 2(7.4)(QR)cos(53°)
Simplifying the equation:
92.16 = 54.76 + QR² - 14.8(QR)cos(53°)
Rearranging the equation:
QR² - 14.8(QR)cos(53°) + 54.76 - 92.16 = 0
QR² - 14.8(QR)cos(53°) - 37.4 = 0
Now we can solve this quadratic equation for QR using the quadratic formula:
QR = [ -(-14.8(cos(53°))) ± √((-14.8(cos(53°)))² - 4(1)(-37.4)) ] / (2(1))
QR = [ 14.8(cos(53°)) ± √((14.8(cos(53°)))² + 149.6) ] / 2
QR = [ 14.8(cos(53°)) ± √(219.04 + 149.6) ] / 2
QR = [ 14.8(cos(53°)) ± √(368.64) ] / 2
QR = [ 14.8(cos(53°)) ± 19.2 ] / 2
Since QR represents length, we take the positive value:
QR = [ 14.8(cos(53°)) + 19.2 ] / 2
QR ≈ 16.13
Now we can use the Law of Sines to find the measure of angle R:
sin(R) / PR = sin(P) / QR
sin(R) = (PR * sin(P)) / QR
sin(R) = (9.6 * sin(53°)) / 16.13
sin(R) ≈ 0.756
R ≈ sin^(-1)(0.756)
R ≈ 48.97°
Therefore, m≤R is approximately 49° to the nearest degree.