Question

A perpendicular bisector, CD, is drawn through point C on AB .
If the coordinates of point A are (-3, 2) and the coordinates of point B are (7, 6), the x-intercept CD of is ___ . Point ___ lies on CD .
Part 1 Options
(3,0)
(18/5,0)
(9,0)
(45/2,0)

Part 2
(-52,141)
(-20,57)
(32,-71)
(54,-128)

Can anyone help me figure this out?

Answer 1

a perpendicular bisector, means, a line that will cut AB segment in two equal segments
so

so, wherever CD cuts AB, is really the MidPoint of AB
since we know what A is, and B is, thus use the midpoint formula
or \(\textit{middle point of 2 points }\\ \quad \\
\begin{array}{lllll}
&x_1&y_1&x_2&y_2\\
% (a,b)
&A({\color{red}{ -3}}\quad ,&{\color{blue}{ 2}})\quad
% (c,d)
&B({\color{red}{ 7}}\quad ,&{\color{blue}{ 6}})
\end{array}\qquad
% coordinates of midpoint
\left(\cfrac{{\color{red}{ x_2}} + {\color{red}{ x_1}}}{2}\quad ,\quad \cfrac{{\color{blue}{ y_2}} + {\color{blue}{ y_1}}}{2} \right)\)

Part #2

now, you'll get some ordered pair, (x,y) for that midpoint
notice, CD cuts AB in half at that point
that means, AB touches that point, and also CD touches that point

CD is perpendicular to AB
so, if you find AB's slope, CD's slope will be THAT, but NEGATIVE RECIPROCAL

then you can find the equation for CD using that slope
\(slope = {\color{green}{ m}} \implies
slope=\cfrac{a}{{\color{blue}{ b}}}\qquad negative\implies -\cfrac{a}{{\color{blue}{ b}}}\qquad reciprocal\implies - \cfrac{{\color{blue}{ b}}}{a}
\\ \quad \\
% point-slope intercept
y-{\color{blue}{ y_1}}={\color{green}{ m}}(x-{\color{red}{ x_1}})\qquad \textit{plug in the values and solve for "y"}\\
\qquad \uparrow\\
\textit{point-slope form}\)


once you know what the equation for CD is
you can test your choices given, to see who lies on CD

Answer 2

so the slope is 4/10?

Answer 3

and the middle point of the two points is (4, 8)?