Question

If $C$ is the midpoint of $A(-1,3)$ and $B(7,3),$ then

• A. $AC=4$ units
• B. $AC=3$ units
• C. Slope of $AC=$ Slope of $CB$
• D. Slope of $AC\ne$ Slope of $CB$

Answer 1

Answer: (A), (C)

Slope of $AC=$ Slope of $CB$

The midpoint of the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by
$(\frac{x_1+y_1}{2},\frac{y_1+y_2}{2})$

$\therefore$ The midpoint of $A(-1,3)$ and $B(7,3)$
$=(\frac{-1+7}{2},\frac{3+3}{2})$
$=(\frac{6}{2},\frac{6}{2})$
$=(3,3)$
So, the coordinates of point $C$ are $(3,3).$

$\therefore A\equiv (-1,3),B\equiv (7,3),C\equiv (3,3)$

Now we have to find the distance between $A$ and $C,$ hence use the distance formula to find the distance of $AC.$
$AC=\sqrt{(3-(-1){)}^2+(3-3{)}^2}=\sqrt{(3+1{)}^2+0^2}=\sqrt{4^2}$
$=4$ units

Now, we have to find the slope of AC and slope of CB to relate them. Using slope formula,
slope of $AC$ $=\frac{3-3}{3-(-1)}$
$=\frac{0}{3+1}=\frac{0}{4}$
$=0$

and slope of $CB=\frac{3-3}{3-7}$
$=\frac{0}{-4}$
$=0$
$\therefore$ Slope of $AC=$ Slope of $CB$
Hence, options (a.) and (c.) are correct choices.