Question

find the equation of the perpendicular bisector of segment ab with endpoints a=2,2 and b=-4,6 if the perpendicular bisector intersects the y-axis at point p what are the lengths of pa and pb

Answer 1

Equation of perpendicular bisector is ,  $y=\frac{3}{2} x+\frac{11}{2}$

Length of both pa and pb are  $\sqrt{\frac{65}{4} }$  unit.

Slope of segment ab is,   $\frac{6-2}{-4-2} =-\frac{2}{3}$

So, slope of perpendicular bisector will be, negative of reciprocal of slope of segment ab.

Therefore, slope of perpendicular bisector is,  $\frac{3}{2}$

Since, perpendicular bisector passes through the mid point of segment ab.

So, coordinate of mid point of segment ab is,

                                             $(\frac{2-4}{2},\frac{2+6}{2} )=(-1,4)$

Thus, perpendicular bisector of ab passing through (- 1, 4) and have slope  3/2.

So, equation of angle bisector is,

                                     $y=\frac{3}{2} x+\frac{11}{2}$

Since, the perpendicular bisector intersects the y-axis at point p

So, coordinate of point p is (0, 11/2)

By applying distance formula,

$pa=\sqrt{(2-0)^{2}+(2-11/2)^{2} }\\\\ =\sqrt{4+\frac{49}{4} }\\\\ =\sqrt{\frac{65}{4} }$

$pb=\sqrt{(-4-0)^{2}+(6-11/2)^{2} }\\\\ =\sqrt{16+\frac{1}{4} }\\\\ =\sqrt{\frac{65}{4} }$

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