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Introduction

Introduction

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The use of the coordinate plane can be helpful with many real-world applications. Scenarios can be translated into equations, equations can be graphed, points can be found, and distances between points can be calculated; but what if you need to find a point on a line that is halfway between two points? Can this be easily done?

6.2.1: Midpoints and Other Points on Line Segments

Lines continue infinitely in both directions. Their length cannot be measured.

A line segment is a part of a line that is noted by two endpoints, (x1, y1) and (x2, y2).

The length of a line segment can be found using the distance formula, .

The midpoint of a line segment is the point on the segment that divides it into two equal parts.

6.2.1: Midpoints and Other Points on Line Segments

Key Concepts, continued

Finding the midpoint of a line segment is like finding the average of the two endpoints.

The midpoint formula is used to find the midpoint of a line segment. The formula is .

You can prove that the midpoint is halfway between the endpoints by calculating the distance from each endpoint to the midpoint.

It is often helpful to plot the segment on a coordinate plane.

DELETE THIS WHILE CHECKING PICKUP: I moved the bottom bullet on this slide from after the table on the next page so things would fit better

6.2.1: Midpoints and Other Points on Line Segments

Key Concepts, continued

Other points, such as a point that is one-fourth the distance from one endpoint of a segment, can be calculated in a similar way.

6.2.1: Midpoints and Other Points on Line Segments

Key Concepts, continued

6.2.1: Midpoints and Other Points on Line Segments

misidentifying x1, x2 and y1, y2

attempting to use the distance formula to locate the midpoint

attempting to use the midpoint formula to find points that split the segment in a ratio other than

using the wrong endpoint to locate the point given a ratio

incorrectly adding or subtracting units from the identified endpoint

6.2.1: Midpoints and Other Points on Line Segments

Guided Practice

Example 1

Calculate the midpoint of the line segment with endpoints (–2, 1) and (4, 10).

Determine the endpoints of the line segment.

The endpoints of the segment are (–2, 1) and (4, 10).

2. Substitute the values of (x1, y1) and (x2, y2) into the midpoint formula and solve.

(1, 5.5)

6.2.1: Midpoints and Other Points on Line Segments

Guided Practice: Example 1, continued

The midpoint of the segment with endpoints (–2, 1) and (4, 10) is (1, 5.5).

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6.2.1: Midpoints and Other Points on Line Segments

Guided Practice

Example 2

Show mathematically that (1, 5.5) is the midpoint of the line segment with endpoints (–2, 1) and (4, 10).

6.2.1: Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued

1. If the ordered pair (1, 5.5) is the midpoint of this segment, then the distance between _____ and _____ should be the same as the distance between _____ and _____.

Distance formula

Substitute (–2, 1) and (1, 5.5).

Simplify as needed.

6.2.1: Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued

The distance between (–2, 1) and (1, 5.5) is units.

6.2.1: Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued

Calculate the distance between the endpoint (4, 10) and the midpoint (1, 5.5).

Use the distance formula.

Distance formula

Substitute (4, 10) and

(1, 5.5).

Simplify as needed.

6.2.1: Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued

The distance between (4, 10) and (1, 5.5) is units. The distance from each endpoint to the midpoint is the same, units, proving that (1, 5.5) is the midpoint of the line segment.

6.2.1: Midpoints and Other Points on Line Segments

Determine the point that is the distance from

the endpoint (–3, 7) of the segment with endpoints(–3, 7) and (5, –9).

6.2.1: Midpoints and Other Points on Line Segments

2. Calculate the distance between the x-values.

|x2 – x1| describes the distance.

|(-3) – 5| describes distance between x-values.

SIMPLIFY.

|-8| = 8.

3. Then, multiply this distance by the given ratio, or ¼.

8 * ¼ = 2, so we know that we will move 2 units horizontally.

6.2.1: Midpoints and Other Points on Line Segments

Repeat this same process with the y-values.

|y2– y1| describes the distance.

|(7) – (-9)| describes distance between y-values.

SIMPLIFY.

|16| = 16.

Then, multiply this distance by the given ratio, or ¼.

16 * ¼ = 4, so we know that we will move 4 units vertically.

6.2.1: Midpoints and Other Points on Line Segments

Because the original question asks us to find the point that is ¼ of the distance from (–3, 7), start at this point and move the number of units that you have found. What point do we end up on?

(-1, 3)

6.2.1: Midpoints and Other Points on Line Segments

6.2.1: Midpoints and Other Points on Line Segments

Example 4

Determine the point that is the distance from the endpoint (2, 9) of the segment with endpoints

(2, 9) and (–4, –6).

6.2.1: Midpoints and Other Points on Line Segments

Endpoints - (2, 9) and (–4, –6).

6.2.1: Midpoints and Other Points on Line Segments

Example 5 – A line segment has one endpoint at (12, 0) and a midpoint of (10, –2). Locate the second endpoint.

1. Determine the endpoints of the line segment.

One endpoint is unknown, this is what we are trying to find!

6.2.1: Midpoints and Other Points on Line Segments

2. Substitute the values of (x1, y1) into the midpoint formula and simplify.

Midpoint formula

Substitute (12, 0)

6.2.1: Midpoints and Other Points on Line Segments

3. Find the value of x.

Create an equation with the known midpoint, (10, -2).

Equation

Simplify

6.2.1: Midpoints and Other Points on Line Segments

3. Find the value of y.

Create an equation to find the value of y.

Equation

Simplify

So the other endpoint of this segment is

(8, -4).

6.2.1: Midpoints and Other Points on Line Segments