Peter (P) and Jamie(J) have computer factories that are 132 miles apart. They both ship their completed parts to Diane (D). Diane is 72 miles from Peter and 84 miles from Jamie. Using the points D,J,and P to form a triangle, find m<PDJ to the nearest tenth of a degree.
Given ΔPDJ,
PJ=132 miles
PD=72 miles
DJ=84 miles
Triangle (angles) can be solved using the cosine rule, since all sides are known.
Let ∠PDJ = θ
Then
cos(θ) = (PD^sup2;+DJ²-PJ²)/(2PD*DJ)
Can you take it from here?
115.4
To find the measure of angle P in triangle PDJ, we need to use the law of cosines. The law of cosines states that for a triangle with sides a, b, and c, and angles A, B, and C opposite those sides, the following formula holds: c^2 = a^2 + b^2 - 2ab*cos(C).
First, let's determine the sides of the triangle PDJ:
- Side DJ = 132 miles (the distance between the computer factories)
- Side DP = 72 miles (the distance between Diane and Peter)
- Side PJ = 84 miles (the distance between Diane and Jamie)
Now we can use the law of cosines to find the measure of angle P:
cos(P) = (DP^2 + PJ^2 - DJ^2) / (2 * DP * PJ)
cos(P) = (72^2 + 84^2 - 132^2) / (2 * 72 * 84)
Calculating the right side of the equation:
cos(P) = (5,184 + 7,056 - 17,424) / (12,096)
cos(P) = (-5,184) / (12,096)
cos(P) = -0.4282
To find the measure of angle P, we need to find the inverse cosine (cos^-1) of -0.4282.
P ≈ cos^-1(-0.4282)
P ≈ 119.9 degrees (rounded to the nearest tenth)
Therefore, the measure of angle PDJ, to the nearest tenth of a degree, is approximately 119.9 degrees.