In triangle PQR, m∠P = 43°, PQ = 7.5, and PR = 8.4 What is m∠R to the nearest degree
Using the Law of Cosines, we have:
QR^2 = PQ^2 + PR^2 - 2PQ*PR*cos(P)
where P is the angle at vertex P. Substituting the given values, we get:
QR^2 = 7.5^2 + 8.4^2 - 2(7.5)(8.4)cos(43°) ≈ 27.901
Taking the square root of both sides, we get:
QR ≈ 5.282
Now, using the Law of Cosines again for angle R, we have:
cos(R) = (PQ^2 + QR^2 - PR^2) / (2PQ*QR)
Substituting the given values, we get:
cos(R) = (7.5^2 + 5.282^2 - 8.4^2) / (2(7.5)(5.282)) ≈ 0.550
Taking the inverse cosine of both sides, we get:
R ≈ 57°
Therefore, m∠R is approximately 57 degrees.
To find the measure of angle R in triangle PQR, we can use the fact that the sum of the angles in a triangle is 180 degrees.
First, we find the measure of angle Q.
Since angle PQR is a straight line (180 degrees), we have:
m∠Q = 180 - m∠P - m∠R
m∠Q = 180 - 43 - m∠R
Next, we substitute the given values into the equation:
m∠Q = 180 - 43 - m∠R
m∠Q = 137 - m∠R
Finally, we solve for m∠R by substituting the values of PQ and PR into the equation:
7.5 + 8.4 + PQ + PR = 180
15.9 + 7.5 + 8.4 = 180
31.8 = 180
Now we can substitute the value of m∠R into the equation for m∠Q:
m∠Q = 137 - m∠R
m∠Q = 137 - 31.8
m∠Q ≈ 105.2 degrees
Therefore, the measure of angle R to the nearest degree is approximately 105 degrees.