# Conjectures 1 - Perpendicular Bisectors

**In an earlier activity, we established the definition of Perpendicular.**

**Definition:**

**Bisect -**to divide into two equal pieces.

**Definition:**

**Perpendicular Bisector**- If a line (or ray or segment) is perpendicular to a segment and also bisect that segment, then it is a

**of that segment.**

*perpendicular bisector***Definition:**

**Equidistant -**of equal distances.

_{1}refers to the distance from C to A while d

_{2}refers to the distance from C to B. Use your spatial intuition in conjunction with these diagrams to answer the questions below.

``

INSTRUCTIONS FOR TASK:**Scenario 1:**If CM is perpendicular to AB, but M in NOT the midpoint of AB, does it seem like d

_{1}and d

_{2}will be equal?

**STEP 1:**Move point C below, make some observations, and answer the question below the applet.

**REFLECT:**
If CM is perpendicular to AB, but M in NOT the midpoint of AB, does it seem like d_{1} and d_{2} will be equal?

**INSTRUCTIONS FOR TASK:**

**Scenario 2:**If M is the midpoint of AB, but CM is NOT perpendicular to AB, does it seem like d

_{1}and d

_{2}will be equal?

**STEP 1:**Move point C below, make some observations, and answer the question below the applet.

**REFLECT:**
If M is the midpoint of AB, but CM is NOT perpendicular to AB, does it seem like d_{1} and d_{2} will be equal?

**INSTRUCTIONS FOR TASK:**

**Scenario 3:**If M is the midpoint of AB and CM is perpendicular to AB, does it seem like d

_{1}and d

_{2}will be equal?

**STEP 1:**Move point C below, make some observations, and answer the question below the applet.

**REFLECT:**
If M is the midpoint of AB and CM is perpendicular to AB, does it seem like d_{1} and d_{2} will be equal?

**REFLECT:**
**Step 2: **In Task 3 above, add another point E on CM (the perpendicular bisector of AB). What do you notice about the distance E is from A and E is from B (in other words how do the lengths of EA and EB compare)? If you move point E along the bisector of AB, does that length relationship change?

Let's formalize this idea. Fill in the blank. Make sure to add this conjecture/theorem to your worksheet.
**Perpendicular Bisector Theorem:**
If a point is on the perpendicular bisector of a segment, the this point is _____________________ from the endpoints of that segment.