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.
(d) Find the sizes of the other two angles of triangle PQR.
(e) Find the length of the median drawn to side PQ.
(f) Find the length of the bisector of angle R.
(g) Find the third side of another triangle that has a 5-inch side, an 8-inch side, and the same area as triangle PQR.
(a) To find the area of triangle PQR, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, segment PR is the base of the triangle, and the height is the length of the segment PQ.
Area = (1/2) * 8 inches * 5 inches
Area = 40 square inches
So, the area of triangle PQR is 40 square inches.
(b) To find the length of the projection of segment PQ onto segment PR, we can use trigonometry. The projection is the length of the perpendicular from point P to segment PR.
We know that the two sides of the triangle are 8 inches and 5 inches, and the angle between them is 60 degrees. Using the sine function, we can find the length of the projection.
Projection = length of PQ * sin(angle)
Projection = 8 inches * sin(60 degrees)
Projection = 8 inches * (√3/2)
Projection = 4√3 inches
So, the length of the projection of segment PQ onto segment PR is 4√3 inches.
(c) To find the length of segment QR, we can use the Law of Cosines. The law states that for any triangle with sides a, b, and c, and angle C opposite side c, we have the following formula:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, segment QR is side c, and segments PQ and PR are sides a and b. The angle opposite side QR is 60 degrees.
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(60 degrees)
QR^2 = 8 inches^2 + 5 inches^2 - 2 * 8 inches * 5 inches * 0.5
QR^2 = 64 + 25 - 80
QR^2 = 9
QR = 3 inches
So, the length of segment QR is 3 inches.
(d) To find the sizes of the other two angles of triangle PQR, we can use the fact that the sum of all angles in a triangle is 180 degrees.
Let's denote the angles as angle Q and angle R.
Angle PQR = 60 degrees (given)
Angle Q + Angle R + 60 degrees = 180 degrees
Angle Q + Angle R = 120 degrees
Since we don't have any additional information about the angles, we cannot determine their exact values. However, we know that the sum of the two angles Q and R is 120 degrees.
(e) To find the length of the median drawn to side PQ, we can use the property of a median in a triangle. A median is a line segment drawn from a vertex of a triangle to the midpoint of the opposite side.
In this case, the median to side PQ starts from vertex R and ends at the midpoint of side PQ. Let's denote the midpoint of PQ as M.
To find the length of the median RM, we can use the following formula:
RM = (1/2) * PQ
RM = (1/2) * 8 inches
RM = 4 inches
So, the length of the median drawn to side PQ is 4 inches.
(f) To find the length of the bisector of angle R, we can use the Angle Bisector Theorem. The theorem states that in a triangle, the angle bisector divides the opposite side into two segments that are proportional to the other two sides.
In this case, the bisector of angle R divides side PQ into two segments. Let's denote the length of the segment PR as x and the length of the segment QR as y.
According to the Angle Bisector Theorem:
PR/QR = PQ/QP
8/x = 5/y
Cross-multiplying, we get:
8y = 5x
We don't have enough information to determine the exact values of x and y, so we cannot find the length of the bisector. However, we know that the relationship between the lengths of the segments PR and QR is given by the equation 8y = 5x.
(g) To find the length of the third side of another triangle that has a 5-inch side, an 8-inch side, and the same area as triangle PQR, we can use the area formula and solve for the missing side.
Let's denote the length of the third side as z.
The area of triangle PQR is 40 square inches (from part a).
Using the area formula:
Area = (1/2) * base * height
40 = (1/2) * z * PQ (Since PQ is the base in this case)
80 = z * 8
z = 10 inches
So, the length of the third side is 10 inches.