The lengths of segments PQ and PR are 8 inches and 5 inches, respectively, and they make a 60-degree angle at P.
(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.
To find the sizes of the other two angles of triangle PQR, we can use the sine rule. The sine rule states that for any triangle ABC, the ratio of a side length to the sine of its opposite angle is constant.
Thus, we can use the sine rule to find the sizes of the other two angles of triangle PQR:
sin(A) / a = sin(B) / b = sin(C) / c
Where A, B, and C are the angles of the triangle, and a, b, and c are the lengths of the opposite sides.
In this case, we know that angle P is 60 degrees and the lengths of segments PQ and PR are 8 inches and 5 inches, respectively.
Using the sine rule, we have:
sin(60) / 8 = sin(A) / 5
Rearranging the equation, we get:
sin(A) = 5 * sin(60) / 8
sin(A) ≈ 0.866 / 8
sin(A) ≈ 0.10825
To find angle A, we can take the inverse sine of 0.10825:
A = arcsin(0.10825)
A ≈ 6.226 degrees
Since the sum of the angles in a triangle is 180 degrees, we can find angle B by subtracting angles P and A from 180:
B = 180 - 60 - 6.226
B ≈ 113.774 degrees
Therefore, the sizes of the other two angles of triangle PQR are approximately 6.226 degrees and 113.774 degrees.
To find the length of the median drawn to side PQ, we can use the median formula, which states that the length of a median drawn from a vertex of a triangle to the midpoint of the opposite side is equal to half the length of that side.
In this case, the median is drawn from vertex R to the midpoint of side PQ. Since side PQ is 8 inches, the length of the median is:
Length of median = 8 / 2
Length of median = 4 inches
Therefore, the length of the median drawn to side PQ is 4 inches.
To find the length of the bisector of angle R, we can use the angle bisector theorem, which states that the length of the bisector of an angle of a triangle is equal to the product of the lengths of the two sides adjacent to the angle, divided by the sum of the lengths of these sides.
In this case, the bisector of angle R divides side PQ into two segments. Since the lengths of segments PQ and PR are 8 inches and 5 inches, respectively, the length of the bisector of angle R is:
Length of bisector = (8 * 5) / (8 + 5)
Length of bisector = 40 / 13
Length of bisector ≈ 3.08 inches
Therefore, the length of the bisector of angle R is approximately 3.08 inches.
To find 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 Heron's formula to find the area of triangle PQR and then use that area to find the length of the third side.
Heron's formula states that the area (A) of a triangle with side lengths a, b, and c is:
A = √(s(s-a)(s-b)(s-c))
Where s is the semi-perimeter of the triangle, given by:
s = (a + b + c) / 2
In this case, we know that the lengths of segments PQ and PR are 8 inches and 5 inches, respectively.
Using Heron's formula, we can calculate the area of triangle PQR:
s = (8 + 5 + c) / 2
s = (13 + c) / 2
A = √((13 + c)(c - 8)(c - 5)((13 + c)/2 - c))
A = √((13 + c)(c - 8)(c - 5)(13 - c)/2)
Since we want the triangle to have the same area as triangle PQR, we can set the area equal to the area of triangle PQR:
√((13 + c)(c - 8)(c - 5)(13 - c)/2) = √(s(s-a)(s-b)(s-c))
Squaring both sides, we get:
((13 + c)(c - 8)(c - 5)(13 - c)/2) = (s(s-a)(s-b)(s-c))
Simplifying and solving for c, we have:
((13 + c)(c - 8)(c - 5)(13 - c)) = 2(13 + c)(c - 5)(c - 8)
Dividing both sides by (13 + c)(c - 8)(c - 5), we get:
13 - c = 2
Solving for c, we find:
c = 11
Therefore, the third side of another triangle that has a 5-inch side, an 8-inch side, and the same area as triangle PQR is 11 inches.
To answer these questions, we can use various geometric principles and formulas. Let's go step by step.
(d) To find the sizes of the other two angles of triangle PQR, we can use the fact that the sum of the angles in a triangle is always 180 degrees. We know that one angle is 60 degrees, so to find the other two angles, we can subtract 60 from 180 and divide the remaining sum by 2.
Angle Q = (180 - 60) / 2 = 60 degrees
Angle R = (180 - 60) / 2 = 60 degrees
Therefore, angles Q and R are both 60 degrees.
(e) To find the length of the median drawn to side PQ, we can use the property that in a triangle, the median from a vertex divides the opposite side into two equal parts. In this case, we need to find the length of the median drawn from vertex R to side PQ.
Since segment QR and segment RP are of equal lengths, the median from R will intersect segment PQ at its midpoint. This means the length of the median is half the length of PQ.
Length of the median = 1/2 * 8 inches = 4 inches
Therefore, 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. According to this theorem, the length of the bisector of an angle in a triangle is proportional to the lengths of the two sides that form the angle.
Let's denote the length of the bisector as x. Then, we can set up the following proportion:
PQ / PR = QX / XR
Substituting the given values:
8 / 5 = x / 8
Cross-multiplying:
40 = 5x
Dividing both sides by 5:
x = 40 / 5 = 8 inches
Therefore, the length of the bisector of angle R is 8 inches.
(g) To find the length of the third side of another triangle with a 5-inch side, an 8-inch side, and the same area as triangle PQR, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
In triangle PQR, the base is 8 inches and the height can be calculated using trigonometry. By splitting the triangle into two right triangles, we can find the height.
Using the sine function:
sin(60 degrees) = height / 5
height = 5 * sin(60 degrees)
height = 5 * sqrt(3) / 2
height = (5 * sqrt(3)) / 2
Area = (1/2) * 8 * ((5 * sqrt(3)) / 2)
Area = 20 * sqrt(3)
Now, for another triangle with the same area, we can use the same formula and solve for the third side (let's call it x):
Area = (1/2) * 5 * x
20 * sqrt(3) = (1/2) * 5 * x
Dividing both sides by (1/2) * 5:
20 * sqrt(3) = x
Therefore, the third side of another triangle with a 5-inch side, an 8-inch side, and the same area as triangle PQR is 20 * sqrt(3) inches.