Chapter 4.7, Problem 15WE

To determine

To prove: if a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

Explanation of Solution

Given information:

Given figure,

Given,

  $AB=AC$

Calculation:

Key steps of proof are as follows:

Suppose  $X\text{is}\text{the}\text{mispoint}\text{of}\overline{BC}$

  $AB=AC$

  $BX=XC\left(X\text{is}\text{the}\text{mispoint}\text{of}\overline{BC}\right)$

  $AX=AX$

  $\Delta AXB\cong \Delta AXC\left(\text{By}\text{SAS}\text{Postulate}\right)$

  $\angle 1\cong \angle 2\left(\text{Corr}\text{.}\angle s\text{of}\cong \Delta s\text{are}\cong \right)$

  $\overline{AX}\perp \overline{BC}\left(\text{If}\text{two}\text{lines}\text{form}\cong \text{adjacent}\angle s,\text{then}\text{the}\text{lines}\text{are}\perp \right)$

$\overline{AX}$ is the perpendicular bisector of $\overline{BC}$

  $\left(X\text{is}\text{the}\text{mispoint}\text{of}\overline{BC}\text{and}\overline{AX}\perp \overline{BC}\right)$