5. 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.
(a) Ah, triangle talk! Time to calculate some areas! To find the area of triangle PQR, we can use the formula (1/2) * base * height. Since we're given the lengths of segments PQ and PR, we can use them as the base and height, respectively. So, the area of triangle PQR is (1/2) * 8 * 5 = 20 square inches. Voila!
(b) Now, let's talk projections. Imagine segment PQ standing in front of a projector and casting its shadow onto segment PR. The length of this projection is the perpendicular distance from P to the line PR. To find this length, we need to use some trigonometry! The projection of PQ onto PR can be calculated as PQ * cos(angle PQR). In this case, angle PQR is 60 degrees. So, the projection length is 8 * cos(60) = 4 inches.
(c) To find the length of segment QR, we turn to the amazing Pythagorean theorem! It states that in a right triangle (which triangle PQR is not), the sum of the squares of the two shorter sides equals the square of the hypotenuse. However, we can create a right triangle by using the projection length as one side. So, we have a right triangle with one leg as the projection length (4 inches), the other leg as segment PR (5 inches), and the hypotenuse as segment QR. Using the Pythagorean theorem, we can find QR: QR^2 = PR^2 - projection length^2 = 5^2 - 4^2 = 25 - 16 = 9. Therefore, QR = √9 = 3 inches.
(d) Now, let's talk angles! We already know that angle PQR is 60 degrees because it was given. So, the sum of the angles in a triangle is 180 degrees. Therefore, the sum of the other two angles must be 180 - 60 = 120 degrees. Since it's not given whether triangle PQR is an isosceles or equilateral triangle, we can't determine the exact sizes of these two angles without more information. But hey, sometimes life is just full of mysteries!
To find the area of triangle PQR, we can use the formula:
Area = (1/2) * base * height
(a) The base of the triangle is PR, which is 5 inches, and the height is the length of the perpendicular from point P to line segment PR. To find the height, we can use the formula:
height = PQ * sin(angle)
Substituting the values given, the height can be calculated as:
height = 8 * sin(60°)
Using the value of sin(60°) = √3/2, we have:
height = 8 * √3/2
= 4√3 inches
Now, substituting the base and height values into the area formula:
Area = (1/2) * 5 * 4√3
= 10√3 square inches
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 formula for the projection of one vector onto another:
Projection = length of one vector * cos(angle)
In this case, the length of vector PQ is 8 inches, and we need to find the projection onto vector PR. The angle between the two vectors is given as 60 degrees.
Using the formula, the length of the projection is:
Projection = 8 * cos(60°)
Using the value of cos(60°) = 1/2, we have:
Projection = 8 * 1/2
= 4 inches
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, which states:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, segment QR is the side opposite angle P in triangle PQR. The lengths of segments PQ and PR are given as 8 inches and 5 inches, respectively, and the angle between them is 60 degrees.
Using the formula, we have:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(60°)
Substituting the given values, and using the value of cos(60°) = 1/2, we have:
QR^2 = 8^2 + 5^2 - 2 * 8 * 5 * 1/2
= 64 + 25 - 80
= 89 - 80
= 9
Taking the square root of both sides, we get:
QR = √9
= 3 inches
Therefore, 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 the angles in any triangle is always 180 degrees.
We already know that angle PQR is 60 degrees. Let's denote the other two angles as α and β.
α + β + 60° = 180°
α + β = 120°
Since we don't have any specific information about the angles, there are infinitely many solutions for α and β that satisfy this equation. Without additional constraints or information, we cannot determine the sizes of the other two angles.
To solve this problem, we will use some geometric properties of triangles. Let's break down each part of the problem and explain the steps to find the answers.
(a) To find the area of triangle PQR, we can use the formula for the area of a triangle, which is given by:
Area = (1/2) * base * height
In this case, segment QR is the base of the triangle, and the height is the perpendicular distance from segment QR to point P. To find the height, we can drop a perpendicular from point P to segment QR and create a right triangle.
We can use the length of segment PR as the base of the right triangle and the length of PR as one of the legs. Since the angle at P is 60 degrees, the other leg will be sin(60) * PR. Therefore, the height (or the length of the perpendicular) is sin(60) * PR.
Once we have the height and the base, we can substitute the values into the formula for the area of a triangle to find the answer.
(b) To find the length of the projection of segment PQ onto segment PR, we can use the concept of projection. The projection of a segment onto another segment is the length of the perpendicular from one endpoint to the other segment.
We can drop a perpendicular from point Q onto segment PR. Let's call the point of intersection between the perpendicular and segment PR as S. The length of the projection of PQ onto PR is the distance between points P and S.
To find this length, we need to determine the length of segment PS. We can use trigonometry to find this length. Since the angle at P is 60 degrees, the length of PS will be cos(60) * PQ.
(c) To find the length of segment QR, we can use the Law of Cosines. The Law of Cosines is a formula used to find the length of a side of a triangle when the lengths of the other two sides and the angle between them are known.
In this case, the lengths of segments PQ and PR and the angle between them (60 degrees) are known. We can use the Law of Cosines to calculate the length of segment QR.
(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. Since we know one angle (60 degrees) and we can find the other two sides and their included angle, we can use the Law of Sines to find the other two angles. The Law of Sines relates the sine of an angle to the length of the side opposite that angle.
By substituting known values into the Law of Sines, we can calculate the sizes of the other two angles of triangle PQR.
Now that we have explained the steps to solve each part of the problem, you can use the appropriate formulas and calculations to find the specific answers.