Answer:
<em>Part a) The </em><em>figure</em><em> a shows that </em><em>AB is parallel to PR</em><em>. Please see attached </em><em>figure a.</em>
<em>Part b) The length of AB is half the length of PR</em>
Step-by-step explanation:
Part A)
Let M be the midpoint formula between points A(x₁, y₁) and B(x₂, y₂)
![M=({\displaystyle {\frac {x_{1}+x_{2}}{2}}, {\frac {y_{1}+y_{2}}{2})](https://tex.z-dn.net/?f=M%3D%28%7B%5Cdisplaystyle%20%7B%5Cfrac%20%7Bx_%7B1%7D%2Bx_%7B2%7D%7D%7B2%7D%7D%2C%20%7B%5Cfrac%20%7By_%7B1%7D%2By_%7B2%7D%7D%7B2%7D%29)
Let A be the midpoint of P(-3, -1) and Q(-1, 7)
The coordinates of the midpoint of P(-3, -1) and Q(-1, 7):
![Mpq{} =({\displaystyle {\frac {-3+(-1)}{2}}, {\frac {-1+7}{2})](https://tex.z-dn.net/?f=Mpq%7B%7D%20%3D%28%7B%5Cdisplaystyle%20%7B%5Cfrac%20%7B-3%2B%28-1%29%7D%7B2%7D%7D%2C%20%7B%5Cfrac%20%7B-1%2B7%7D%7B2%7D%29)
![Mpq{} =(-2, 3)](https://tex.z-dn.net/?f=Mpq%7B%7D%20%3D%28-2%2C%203%29)
Let B be the midpoint of Q(-1, 7) and R(3, 3):
![Mqr{} =({\displaystyle {\frac {-1+(3)}{2}}, {\frac {7+3}{2})](https://tex.z-dn.net/?f=Mqr%7B%7D%20%3D%28%7B%5Cdisplaystyle%20%7B%5Cfrac%20%7B-1%2B%283%29%7D%7B2%7D%7D%2C%20%7B%5Cfrac%20%7B7%2B3%7D%7B2%7D%29)
![Mqr{} =(1, 5)](https://tex.z-dn.net/?f=Mqr%7B%7D%20%3D%281%2C%205%29)
The <em>figure</em> a shows that <em>AB is parallel to PR</em>. Please see attached <em>figure a</em>.
Part B)
The point A has the coordinates: A(-2, 3)
The point B has the coordinates: B(1, 5)
The distance formula is: ![d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}](https://tex.z-dn.net/?f=d%3D%7B%5Csqrt%20%7B%28x_%7B2%7D-x_%7B1%7D%29%5E%7B2%7D%2B%28y_%7B2%7D-y_%7B1%7D%29%5E%7B2%7D)
So, the length of AB ![={\sqrt {(1-(-2))^{2}+(5-3)^{2}](https://tex.z-dn.net/?f=%3D%7B%5Csqrt%20%7B%281-%28-2%29%29%5E%7B2%7D%2B%285-3%29%5E%7B2%7D)
= ![\sqrt{9+4}](https://tex.z-dn.net/?f=%5Csqrt%7B9%2B4%7D)
= ![\sqrt{13}](https://tex.z-dn.net/?f=%5Csqrt%7B13%7D)
= ![3.6056](https://tex.z-dn.net/?f=3.6056)
And, the length of PR ![={\sqrt {(3-(-3))^{2}+(3-(-1)^{2}](https://tex.z-dn.net/?f=%3D%7B%5Csqrt%20%7B%283-%28-3%29%29%5E%7B2%7D%2B%283-%28-1%29%5E%7B2%7D)
= ![\sqrt{36+16}](https://tex.z-dn.net/?f=%5Csqrt%7B36%2B16%7D)
= ![\sqrt{52}](https://tex.z-dn.net/?f=%5Csqrt%7B52%7D)
= ![2\sqrt{13}](https://tex.z-dn.net/?f=2%5Csqrt%7B13%7D)
= = ![7.2111](https://tex.z-dn.net/?f=7.2111)
As the length of AB = 3.6056 and the length of PR = 7.2111
<em>So, it is clear that the length of AB is half the length of PR.</em>
<em>Keywords: distance formula, midpoints, triangle, coordinate plane</em>
<em>Learn more about midpoints from brainly.com/question/7560674</em>
<em>#learnwithBrainly</em>