Perimeter of the Quadrilateral ARPQ in Triangle ABC

Given:
AC = 21 cm
BC = 29 cm
AB = 30 cm

To find the perimeter of the quadrilateral ARPQ, we need to find the lengths of the sides AP, PQ, QR, and RA.

Step 1: Finding the length of AP

Since P is the midpoint of BC, we can use the midpoint formula to find the coordinates of point P. Let's assume A(0,0), B(x1,y1), and C(x2,y2).

Coordinates of P = ((x1 + x2)/2, (y1 + y2)/2)

Let's find the coordinates of point P. Since P is the midpoint of BC, the coordinates of B and C will be (x1, y1) and (x2, y2) respectively.

Coordinates of P = ((x2 + 0)/2, (y2 + 0)/2)
= (x2/2, y2/2)

Since P is the midpoint of BC, we can say that P is the midpoint of the line segment BC. Therefore, the coordinates of P will be the average of the coordinates of B and C.

Now, let's find the coordinates of B and C. We know that B is (x1,y1) and C is (x2,y2).

Coordinates of B = (x1, y1)
Coordinates of C = (x2, y2)

From the given information, we know that AC = 21 cm, BC = 29 cm, and AB = 30 cm. Using the distance formula, we can find the values of x1, y1, x2, and y2.

Distance between A and C = AC = √((x2 - 0)^2 + (y2 - 0)^2) = 21 cm
Distance between B and C = BC = √((x2 - x1)^2 + (y2 - y1)^2) = 29 cm
Distance between A and B = AB = √((x1 - 0)^2 + (y1 - 0)^2) = 30 cm

Solving these equations will give us the values of x1, y1, x2, and y2.

Step 2: Finding the length of PQ, QR, and RA

Since Q is the midpoint of CA, and R is the midpoint of AB, we can use the midpoint formula to find their coordinates and then calculate their distances.

Coordinates of Q = ((0 + x2)/2, (0 + y2)/2) = (x2/2, y2/2)
Coordinates of R = ((x1 + 0)/2, (y1 + 0)/2) = (x1/2, y1/2)

Using the distance formula, we can find the lengths of PQ, QR, and RA.

Distance between P and Q = √((x2/2 - 0)^2 + (y2/2 - 0)^2)
Distance between Q and R = √((x1/2 - x2/2)^2 + (y1/2 - y2/2)^2)
Distance between R and A = √((x1/