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ACT Prep: Help and Review44 chapters | 435 lessons | 26 flashcard sets

Instructor:
*Beverly Maitland-Frett*

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

This lesson will discuss the concepts of equidistant and midpoint. We will explore the midpoint formula and show some examples of relevance in the real world setting.

In soccer, the midfielder is a highly regarded position. If you are a soccer fan, you will agree that this player is posted somewhere around the middle of the field, not at either goals. Likewise in American football, the midfield is an important position on the field. Equidistant, in concept, may be associated with these players and positions, in that they all have to do with the middle. Equidistant comes from two words, equal and distant. An **equidistant** point is a point that is an equal distance from two other points. Please note, an equidistant point is not necessarily in the middle of two points. For example, you may live near two Walmart locations that are both 6.5 miles away from you. However, your house is not necessarily located between the two Walmart locations.

Therefore, any point that is equidistant may be a midpoint or it may also be points that lie on the **perpendicular bisector** of a line segment. Line segment RS as shown below, is the perpendicular bisector of line segment PQ.

Remember that the **perpendicular bisector** is the line or line segment that cuts another line in half, simultaneously forming a right angle. Look at the diagram below, since points R, O, N, M, T, S all lie on the perpendicular bisector, they are equidistant from points P and Q. However, only point M is the midpoint of line PQ.

If a pilot, sailor or a voyager wants to know about location and distances, they will use a map to investigate distances. Maps provide exact location and coordinates of places. Sometimes we look at maps and guess how far away a place is with respect to our present location. In order to find a point that is equidistant from other points, we can use the midpoint formula, once we know the *x* and *y* coordinates.

We add both *x* coordinates and divide by two, then add both the *y* coordinates and divide by two.

**Example 1**

Point E is the midpoint of line segment LM. Point E is equidistant from Point L and Point M.

**Example 2**

Given coordinates A (-3, 11) and B (4, -8), what are the coordinates for the midpoint of line segment AB?

(-3 + 11)/2, (4 + -8)/2

(8/2, -4/2)

(4, -2)

Our midpoint is (4, -2).

- A point that is
**equidistant**is the same distance away from two other points. - An equidistant point is not necessarily in the middle of the other two comparable points. Therefore, an equidistant point may or may not be a midpoint.
- All points on the perpendicular bisector of a line segment are equidistant from the endpoints of that line segment.
- To find a point that is equidistant, we can use the midpoint formula:

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ACT Prep: Help and Review44 chapters | 435 lessons | 26 flashcard sets

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