Question

activity in this activity, you will use the geogebra to find a relationship between points on a perpendicular bisector and the endpoints of the line segment it bisects. open perpendicular bisectors, and complete each step below. for help, watch these short videos about using geogebra to measure and create points, lines, and angles. part a in the diagram, \\( \overleftrightarrow{cd} \perp \overline{ab} \\). use the geogebra tools to draw \\( \overline{ac} \\) and \\( \overline{bc} \\) and measure their lengths. how do the lengths compare?

Explanation:

Step1: Recall perpendicular bisector property

A perpendicular bisector of a segment is a line that is perpendicular to the segment and passes through its midpoint, meaning it divides the segment into two equal parts. For any point $C$ on the perpendicular bisector $\overleftrightarrow{CD}$ of $\overline{AB}$, the distances from $C$ to the endpoints $A$ and $B$ are equal.

Step2: Apply property to lengths

By the perpendicular bisector theorem, for point $C$ on $\overleftrightarrow{CD}$ (where $\overleftrightarrow{CD} \perp \overline{AB}$ and $\overleftrightarrow{CD}$ bisects $\overline{AB}$):
$\overline{AC} = \overline{BC}$

Answer:

The lengths of $\overline{AC}$ and $\overline{BC}$ are equal.