To answer the given questions you need to find the equation of the line CD.
Equation of a line (y=mx+b)
1. As the bisector CD is perpendicular to AB; if AB has a slope m, CD has a slope -1/m (opposite of the reciprocal)
Use the coordinates of points A and B to find the slope (m) of AB:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex][tex]m=\frac{6-2}{7-(-3)}=\frac{4}{7+3}=\frac{4}{10}=\frac{2}{5}[/tex]
Then, the slope of CD is:
[tex]-\frac{1}{m}=-\frac{1}{\frac{2}{5}}=-\frac{5}{2}[/tex]
2. To find the y-intercept (b); find a point in CD.
As CD is a bisector of AB, it passes through the midpoint of AB; use the next formula to find the midpoint:
[tex]\begin{gathered} M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \\ \\ M=(\frac{-3+7}{2},\frac{2+6}{2}) \\ \\ M=(\frac{4}{2},\frac{8}{2}) \\ \\ M=(2,4) \end{gathered}[/tex]
The line CD passes through the point (2,4), use it and the slope (-5/2) to find the y-intercept as follows:
[tex]\begin{gathered} y=mx+b \\ 4=-\frac{5}{2}(2)+b \\ \\ 4=-5+b \\ 4+5=b \\ 9=b \end{gathered}[/tex]
3. Write the equation of the line CD: us the slope and y-intercept above
m=-5/2
b=9
[tex]y=-\frac{5}{2}x+9[/tex]
4. To find the x-intercept solve x using the equation of the line when y=0:
[tex]\begin{gathered} 0=-\frac{5}{2}x+9 \\ -9=-\frac{5}{2}x \\ \\ 2(-9)=-5x \\ -18=-5x \\ \frac{-18}{-5}=x \\ \\ x=\frac{18}{5} \end{gathered}[/tex]X-intercept has the coordinates: (18/5,0)
5. Use the equation of the line and evaluate it for each of the given points to find if the line passes through it:
[tex]\begin{gathered} (-52,141) \\ 141=-\frac{5}{2}(-52)+9 \\ 141=130+9 \\ 141\ne139 \\ \\ (-20,57) \\ 57=-\frac{5}{2}(-20)+9 \\ 57=50+9 \\ 57\ne59 \\ \\ (32,-71) \\ -71=-\frac{5}{2}(32)+9 \\ -71=-80+9 \\ -71=-71 \\ \\ (54,-128) \\ -128=-\frac{5}{2}(54)+9 \\ -128=-135+9 \\ -128\ne-126 \end{gathered}[/tex]
As you can see above the point that makes a truth math equation is (32,-71)
Point (32,-71) lies on CD
