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

Some useful formulas to help you:

Let (x1,y1) and (x2,y2) be two given points

- Mid-point
between two given points is ((x1+x2)/2,(y1+y2)/2).

- length, L
between two points: L=sqrt((y2-y1)²+(x2-x1)²)

To calculate the lengths of the midsegments in triangle PQR, we need to find the midpoints of each side of the triangle. The midpoint of a line segment is the average of the x-coordinates and the average of the y-coordinates.

a. Calculation of midsegment lengths:
1. PQ midsegment: The midpoint of PQ is ((7 + (-3))/2, (7 + (-5))/2) = (2, 1). The length of the PQ midsegment can be found by using the distance formula between the midpoint and the third vertex R. Let's call the midpoint of PQ as M1.
Length of PQ midsegment = √((5 - 2)^2 + (-3 - 1)^2) = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5.

2. QR midsegment: The midpoint of QR is ((-3 + 5)/2, (-5 + (-3))/2) = (1, -4). The length of the QR midsegment can be found by using the distance formula between the midpoint and the first vertex P. Let's call the midpoint of QR as M2.
Length of QR midsegment = √((7 - 1)^2 + (7 - (-4))^2) = √(6^2 + 11^2) = √(36 + 121) = √157.

3. RP midsegment: The midpoint of RP is ((5 + 7)/2, (-3 + 7)/2) = (6, 2). The length of the RP midsegment can be found by using the distance formula between the midpoint and the second vertex Q. Let's call the midpoint of RP as M3.
Length of RP midsegment = √((-3 - 6)^2 + (-5 - 2)^2) = √((-9)^2 + (-7)^2) = √(81 + 49) = √130.

b. Calculation of side lengths of triangle PQR:
1. Length of PQ: Using the distance formula between the points P and Q:
Length of PQ = √((7 - (-3))^2 + (7 - (-5))^2) = √(10^2 + 12^2) = √(100 + 144) = √244.

2. Length of QR: Using the distance formula between the points Q and R:
Length of QR = √((-3 - 5)^2 + (-5 - (-3))^2) = √((-8)^2 + (-2)^2) = √(64 + 4) = √68.

3. Length of RP: Using the distance formula between the points R and P:
Length of RP = √((5 - 7)^2 + (-3 - 7)^2) = √((-2)^2 + (-10)^2) = √(4 + 100) = √104.

c. Comparison between midsegment lengths and side lengths:
If we compare the length of each midsegment with the corresponding side length of the triangle, we notice that the midsegment lengths are always half the length of the corresponding side lengths.
For example, the length of PQ midsegment (5) is half the length of PQ (2√44 ≈ 9.8), the length of QR midsegment (√157) is half the length of QR (√68 ≈ 8.25), and the length of RP midsegment (√130) is half the length of RP (√104 ≈ 10.2).
This relationship holds true for any triangle, not just the given one.

use distance formula to get distance (length of sides) between two points:

length = sqrt((y2-y1)^2+(x2-x1)^2)
use midpoint formula to get midpoint of one side/length:
midpoint = ((x1+x2)/2,(y1+y2)/2)
*for sure, the answer in (b) is twice the answer in (a) [of course with respect to the same side]

so there,, :)