The lengths of segments PQ and PR are 8 inches and 5 inches, respectively, and they make a 60-degree angle at P.
(a) Find the area of triangle PQR.
(b) Find the length of the projection of segment PQ onto segment PR.
(c) Find the length of segment QR.
To solve these problems, we can apply trigonometric principles such as the Law of Cosines and the Law of Sines. Let's break down each question step by step.
(a) To find the area of triangle PQR, we can use the formula for the area of a triangle given two sides and the angle between them. In this case, we have the lengths of segments PQ and PR, as well as the angle between them, which is 60 degrees.
The formula for the area of a triangle given two sides and the included angle is:
Area = (1/2) * PQ * PR * sin(angle)
Substituting the given values into the formula, we have:
Area = (1/2) * 8 in * 5 in * sin(60 degrees)
To evaluate sin(60 degrees), we can refer to a trigonometric table or use a calculator. The value of sin(60 degrees) is √3/2.
Plugging in the values, we have:
Area = (1/2) * 8 in * 5 in * (√3/2)
Area = 20 * (√3/2)
Area = 10√3
Therefore, the area of triangle PQR is 10√3 square inches.
(b) To find the length of the projection of segment PQ onto segment PR, we can use the Law of Cosines. The projection length can be found using the formula:
Projection Length = PQ * cos(angle)
In this case, the angle is 60 degrees. Substituting the values into the formula, we have:
Projection Length = 8 in * cos(60 degrees)
To evaluate cos(60 degrees), we can refer to a trigonometric table or use a calculator. The value of cos(60 degrees) is 1/2.
Plugging in the values, we have:
Projection Length = 8 in * (1/2)
Projection Length = 4 in
Therefore, the length of the projection of segment PQ onto segment PR is 4 inches.
(c) To find the length of segment QR, we can use the Law of Cosines. The law states that in any 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 adjacent sides and the cosine of the included angle.
The formula for the Law of Cosines is:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(angle)
Substituting the given values, we have:
QR^2 = 8 in^2 + 5 in^2 - 2 * 8 in * 5 in * cos(60 degrees)
To evaluate cos(60 degrees), we can refer to a trigonometric table or use a calculator. The value of cos(60 degrees) is 1/2.
Plugging in the values, we have:
QR^2 = 64 in^2 + 25 in^2 - 2 * 8 in * 5 in * (1/2)
QR^2 = 89 in^2 - 40 in^2
QR^2 = 49 in^2
QR = √49 in
QR = 7 in
Therefore, the length of segment QR is 7 inches.