The lengths of segments PQ and PR are 8 inches and 5 inches, respectively, and they make a 60-degree angle at P.

(1) Find the sizes of the other two angles of triangle PQR.
(2) Find the length of the median drawn to side PQ.
(3) Find the length of the bisector of angle R.
(4) 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 answer each of these questions, we need to apply various principles of triangle geometry. Let's go step by step:

(1) To find the sizes of the other two angles of triangle PQR, we can use the fact that the sum of the angles of any triangle is 180 degrees. We already know that angle P is 60 degrees. Let's call the other two angles Q and R.

Sum of angles PQR + Q + R = 180 degrees
60 degrees + Q + R = 180 degrees

Since we have a triangle, we know that the sum of angles Q and R must be 180 - 60 = 120 degrees. However, we don't have enough information to determine the exact values of Q and R because we don't know the relationship between the sides PQ and PR.

(2) To find the length of the median drawn to side PQ, we need to use the concept that a median of a triangle divides the opposite side into two equal segments. Let's denote the midpoint of side PQ as M and the length of the median as x.

Using the properties of medians, we can say that PM = MQ = x. Therefore, we have two identical right triangles formed, PMR and PQR. The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides for any right triangle.

In triangle PMR:
PR^2 = PM^2 + MR^2
5^2 = x^2 + MR^2
25 = x^2 + MR^2

In triangle PQR:
PQ^2 = QR^2 + PR^2
8^2 = QR^2 + 5^2
64 = QR^2 + 25
QR^2 = 64 - 25
QR^2 = 39

Since PMR and PQR are congruent, MR = QR. Substituting QR^2 = 39:
MR^2 = 39

Now we can substitute MR^2 into the equation for PMR:
25 = x^2 + MR^2
25 = x^2 + 39
x^2 = 25 - 39
x^2 = -14

However, we cannot have a negative length, so there seems to be an error or missing information in the given problem.

(3) To find the length of the bisector of angle R, we can use the angle bisector theorem, which states that in any triangle, the ratio of the lengths of two sides divided by their corresponding angle bisector segments is equal.

Let's call the length of the bisector segment that intersects side PQ as m. According to the theorem:

PR / PQ = MR / MQ

Substituting the given lengths:
5 / 8 = MR / MQ

Now we can solve for MR using the fact that we know MQ = x from part (2):
5 / 8 = MR / x

Cross-multiplying,
5x = 8MR

Next, we need to substitute MR using the triangle PQR relationship MR = QR (as established in part (2)):
5x = 8QR

Again, we don't have the length value of QR, so we cannot determine the length of the bisector.

(4) 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 the formula for the area of a triangle:

Area = (1/2) * base * height

Given that PQR is a right triangle with a 60-degree angle at P, we can see that the segment PQ is the base and PR is the height. The formula for the area of a triangle reduces to:

Area = (1/2) * PQ * PR

Plugging in the given lengths:
Area = (1/2) * 8 * 5
Area = 20 square inches

To find the third side, let's call it RS, we use the formula for the area of a triangle given three sides, also known as Heron's formula:

Area = sqrt(s * (s - a) * (s - b) * (s - c))

Where s is the semiperimeter (half of the sum of the triangle's three sides) and a, b, and c are the lengths of the triangle's sides.

We know two sides, 5 inches (PR) and 8 inches (PQ). The length of the third side is RS. The semiperimeter is (PR + PQ + RS) / 2.

Using the area formula and the given area (20) and the lengths of PR (5) and PQ (8), we can calculate:

20 = sqrt((5 + 8 + RS) / 2 * (RS / 2) * (RS / 2 - 5) * (RS / 2 - 8))

We can solve this equation to find the length of RS using algebraic methods or numerical approximation techniques.

Please note that the given values may lead to non-real solutions or there may be missing or incorrect information in the problem.