Answer:
<em>See the proof below</em>
Step-by-step explanation:
Given the following coordinates
P(2, −1)
Midpoint of PQ M(3, 0)
We can get the coordinate point Q using the midpoint formula;
M(X,Y) = (x1+x2/2, y1+y2/2)
X = x1+x2/2
3 = 2+x2/2
6 = 2+x2
x2 = 6-2
x2 = 4
Y = y1+y2/2
0 = -1+y2/2
0 = -1 + y2
y2 = 0+1
y2 = 1
<em>Hence the coordinate of Q is (4, 1)</em>
Next is to get the coordinate of R
Given the midpoint of QR to be N(5, 3)
(5,3) = (4+x2/2, 1+y2/2)
5 = 4+x2/2
10 = 4+x2
x2 = 10-4
x2 = 6
1+y2/2 = 3
1+y2 = 6
y2 = 6-1
y2 = 5
<em>Hence the coordinate of R is (6,5)</em>
<em></em>
Given the coordinates M(3, 0) and N(5, 3)
Slope is expressed as:
m = y2-y1/x2-x1
m = 3-0/5-3
m = 3/2
Slope of MN = 3/2
Get the slope of PR
Given the coordinates P(2, −1) and R (6,5)
Slope of PR = 5-(-1)/6-2
Slope of PR = 5+1/4
Slope of PR = 6/4 = 3/2
<em>Since the slope of MN is equal to that of PR, hence MN is parallel to PR i.e MN || PR</em>
<em></em>
To show that MN = 1/2PR, we will have to take the distance between M and N and also P and R first as shown:
For MN with coordinates M(3, 0) and N(5, 3)
MN = √(x2-x1)²+(y2-y1)²
MN = √(5-3)²+(3-0)²
MN = √2²+3²
MN = √13
Get the length of PR where P(2, −1) and R (6,5)
PR = √(6-2)²+(5+1)²
PR = √4²+6²
PR = √16+36
PR = √52
PR = √4*13
PR = √4*√13
PR = 2√13
Since MN = √13
PR = 2MN
Divide both sides by 2
PR/2 = 2MN/2
PR/2 = MN
Hence MN = 1/2 PR (Proved!)
= 10