Given:
Endpoints of a line segment AB are A(2,3) and B(8,11).
To find:
(1) Slope of AB.
(2) Length of AB.
(3) Midpoint of AB.
(4) Slope of a line perpendicular to AB.
Solution:
We have, endpoints of line segment AB, A(2,3) and B(8,11).
(1)
Slope of AB is
![m=\dfrac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=m%3D%5Cdfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
![m=\dfrac{11-3}{8-2}](https://tex.z-dn.net/?f=m%3D%5Cdfrac%7B11-3%7D%7B8-2%7D)
![m=\dfrac{8}{6}](https://tex.z-dn.net/?f=m%3D%5Cdfrac%7B8%7D%7B6%7D)
![m=\dfrac{4}{3}](https://tex.z-dn.net/?f=m%3D%5Cdfrac%7B4%7D%7B3%7D)
Therefore, the slope of AB is
.
(2)
Length of AB is
![AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=AB%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
![AB=\sqrt{(8-2)^2+(11-3)^2}](https://tex.z-dn.net/?f=AB%3D%5Csqrt%7B%288-2%29%5E2%2B%2811-3%29%5E2%7D)
![AB=\sqrt{(6)^2+(8)^2}](https://tex.z-dn.net/?f=AB%3D%5Csqrt%7B%286%29%5E2%2B%288%29%5E2%7D)
![AB=\sqrt{36+64}](https://tex.z-dn.net/?f=AB%3D%5Csqrt%7B36%2B64%7D)
![AB=\sqrt{100}](https://tex.z-dn.net/?f=AB%3D%5Csqrt%7B100%7D)
![AB=10](https://tex.z-dn.net/?f=AB%3D10)
Therefore, the length of AB is 10 units.
(3) Midpoint of AB is
![Midpoint=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)](https://tex.z-dn.net/?f=Midpoint%3D%5Cleft%28%5Cdfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%5Cdfrac%7By_1%2By_2%7D%7B2%7D%5Cright%29)
![Midpoint=\left(\dfrac{2+8}{2},\dfrac{3+11}{2}\right)](https://tex.z-dn.net/?f=Midpoint%3D%5Cleft%28%5Cdfrac%7B2%2B8%7D%7B2%7D%2C%5Cdfrac%7B3%2B11%7D%7B2%7D%5Cright%29)
![Midpoint=\left(\dfrac{10}{2},\dfrac{14}{2}\right)](https://tex.z-dn.net/?f=Midpoint%3D%5Cleft%28%5Cdfrac%7B10%7D%7B2%7D%2C%5Cdfrac%7B14%7D%7B2%7D%5Cright%29)
![Midpoint=\left(5,7\right)](https://tex.z-dn.net/?f=Midpoint%3D%5Cleft%285%2C7%5Cright%29)
Therefore, the midpoint of AB is (5,7).
(4)
Product of slopes of two perpendicular lines is -1.
Let the slope of line perpendicular to AB be m₁.
![m_1\times \dfrac{4}{3}=-1](https://tex.z-dn.net/?f=m_1%5Ctimes%20%5Cdfrac%7B4%7D%7B3%7D%3D-1)
![m_1=-\dfrac{3}{4}](https://tex.z-dn.net/?f=m_1%3D-%5Cdfrac%7B3%7D%7B4%7D)
So, slope of line perpendicular to AB is
.