A triangle has vertices P(7,7), Q(-3,-5), and R(5,-3).

a. calculate the lengths of the midsegmaents
b. calculate the lengths of the three side of triangle PQR.
c. compare your answers in a. and b. what do you notice?

P(7 , 7) , Po(2 , 1) , Q(-3 , -5).

Q(-3 , -5) , Qo(1 , -4) , R(5 , -3).

R(5 , -3) , Ro(6 , 2) , P(7 , 7).

a. (PPo)^2 = (2 - 7)^2 + (1 - 7)^2,
= 25 + 36 = 61,
PPo = sqrt(61) = 7.8.

(QQo)^2 = (1 - (-3))^2 + (-4 -(-5))^2,
= 16 + 1 = 17,
QQo = sqrt(17) = 4.1.

(RRo)^2 = (6 - 5)^2 + (2 - (-3))^2,
= 1 + 25 = 26,
RRo = sqrt(26) = 5.1.

b. (PQ)^2 = (-3 -7)^2 + (-5 - 7)^2,
= 100 + 144 = 244,
PQ = sqrt(244) = 15.6.

(QR)^2 = (5 - (-3))^2 + (-3 -(-5))^2,
= 64 + 4 = 68,
QR = sqrt(68) = 8.2.

(RP)^2 = (7 - 5)^2 + (7 - (-3))^2,
= 4 + 100 = 104,
RP = sqrt(104) = 10.2.

The Mid-Point formula was used to calculate Po , Qo , Ro:

Xo = (x1 + x2) /2.
Yo = (y1 + y2) / 2.

To calculate the lengths of the midsegments of a triangle, we need to find the midpoints of the sides and then calculate the distance between those midpoints. Let's follow these steps:

a. Calculate the lengths of the midsegments:
1. Find the midpoints of the sides using the formula:
Midpoint = [(x1 + x2) / 2, (y1 + y2) / 2]

Midpoint of PQ: [((7 + (-3)) / 2), ((7 + (-5)) / 2)] = [2, 1]
Midpoint of QR: [((-3 + 5) / 2), ((-5 + (-3)) / 2)] = [1, -4]
Midpoint of RP: [((5 + 7) / 2), ((-3 + 7) / 2)] = [6, 2]

2. Calculate the lengths of the midsegments using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Length of midsegment PQ: sqrt((2 - 1)^2 + (1 - (-4))^2) = sqrt(1 + 25) = sqrt(26)
Length of midsegment QR: sqrt((1 - 6)^2 + (-4 - 2)^2) = sqrt(25 + 36) = sqrt(61)
Length of midsegment RP: sqrt((6 - 2)^2 + (2 - 1)^2) = sqrt(16 + 1) = sqrt(17)

b. Calculate the lengths of the three sides of triangle PQR:
1. Calculate the distance between the given vertices using the distance formula:
Length of side PQ: sqrt((7 - (-3))^2 + (7 - (-5))^2) = sqrt(100 + 144) = sqrt(244)
Length of side QR: sqrt((-3 - 5)^2 + (-5 - (-3))^2) = sqrt(64 + 4) = sqrt(68)
Length of side RP: sqrt((5 - 7)^2 + (-3 - 7)^2) = sqrt(4 + 100) = sqrt(104)

c. Comparing the answers in a. and b., we notice that the lengths of the midsegments (PQ, QR, and RP) are equal to half the lengths of the sides (PQ, QR, and RP) of the triangle PQR. In other words, the midsegments are half the length of the corresponding sides.

This property holds true for any triangle, not just this specific example.