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# 6.1: Perpendicular Bisector

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 5, Lesson 2.

## Problem 1 – Exploring the Perpendicular Bisector Theorem

Start the Cabri Jr. application by pressing APPS and selecting CabriJr. Open the file PERBIS by pressing Y=\begin{align*}Y=\end{align*}, selecting Open..., and selecting the file.

Line CD\begin{align*}CD\end{align*} is the perpendicular bisector of AB¯¯¯¯¯¯¯¯\begin{align*}\overline{AB}\end{align*}. Find AC\begin{align*}AC\end{align*} and BC\begin{align*}BC\end{align*} using the Distance and Length tool (press GRAPH and select Measure > D.&Length). Remember that AC\begin{align*}AC\end{align*} means the length of AC¯¯¯¯¯¯¯¯\begin{align*}\overline{AC}\end{align*}.

1. Move point C\begin{align*}C\end{align*} to 4 different positions and record the measurements in the table below. To move the point, move the cursor over the point, press a\begin{align*}a\end{align*}, move the point to the desired location, and then press a\begin{align*}a\end{align*} again to release the point.

Position 1st\begin{align*}1^{st}\end{align*} position 2nd\begin{align*}2^{nd}\end{align*} position 3rd\begin{align*}3^{rd}\end{align*} position 4th\begin{align*}4^{th}\end{align*} position
AC\begin{align*}AC\end{align*}
\begin{align*}BC\end{align*}

2. What is the relationship between the measurements of \begin{align*}AC\end{align*} and \begin{align*}BC\end{align*}?

3. Make a conjecture based on your results above about a point on the perpendicular bisector and the endpoints of a segment.

## Problem 2 – An Application of the Perpendicular Bisector Theorem

John and Jane are a young college graduate couple who are relocating to a new city. They have jobs at separate locations, but work out at the same gym. They would like to buy a house that is equidistant from their jobs and gym. They use the map below and see that John’s workplace is located at \begin{align*}B7\end{align*}, Jane’s workplace is at \begin{align*}J6\end{align*}, and their gym is located at \begin{align*}E3\end{align*}.

Open the Cabri Jr. file POINTS. Three points are plotted: ordered pair (1, 3.5) represents \begin{align*}B7\end{align*}; ordered pair (5, 3) represents \begin{align*}J6\end{align*}; and ordered pair (2.5, 1.5) represents \begin{align*}E3\end{align*}.

1. Use Perpendicular Bisector Theorem to decide where John and Jane should live.
2. Using the map’s notation, where should the young couple live?

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