Mrs. McConaughy Geometry 1

The Coordinate Plane

During this lesson you will:

Find the distance between two points in the plane Find the coordinates of the midpoint of a segment

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PART I: FINDING DISTANCE

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The Coordinate PlaneQuadrant I (+, +) Quadrant II (-,

+)

Quadrant III (-, -) Quadrant IV (+, -)

(0,0)

TThe coordinates of point T are ________.(6,3)

Origin

The Coordinate Plane

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When working with Coordinate Geometry, there are many ways to

find distances (lengths) of line segments on graph paper. Let's

examine some of the possibilities:

Method 1:Whenever the segments are horizontalor vertical, the length can be obtained

by counting.  

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Method One

Whenever the segments are horizontal or vertical, the length can be obtained by counting.

When we need to find the length (distance) of a segment such as AB, we simply COUNT the distance from point A to point B.(AB = ___)

We can use this same counting approach for CD .(CD = ___)

Unfortunately, this counting approach does NOT work for EF which is a diagonal segment.

7

3

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Method 2: To find the distance between two points, A(x1, y1) and

B(x2, y2), that are not on a horizontal or vertical line, we can

use the Distance Formula.

Formula

The Distance Formula

The distance, d, between two points,

A(x1, y1) and B(x2, y2), is Alert! The Distance Formula can be used forall line segments: vertical, horizontal, and diagonal.

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Finding Distance

What is the distance between the two points on the right?

STEP 1: Find the coordinates of the two points.____________

STEP 2: Substitute into the Distance Formula.

(0,0)

(6,8)

(0,0) (6,8)

ALERT! Order is important when using Distance Formula.

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Example: Given (0,0) and (6,8), find the distance between the two points.

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Applying the Distance Formula

(2,4)Jackson

(__,__) Symphony(__,__) City Plaza(__,__) Cedar

(__,__) Central

(__,__) North

(__,__) Oak

Each morning H. I. Achiever takes the “bus line” from Oak to Symphony. How far is the bus ride from Oak to Symphony?

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Final Checks for Understanding

1. State the Distance Formula in words.

2. When should the Distance Formula be used when determining the distance between two given points?

3. Find the length of segment AB given A (-1,-2) and B (2,4).

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Homework Assignment

Page 46, text: 1-17 odd.

*Extra Practice WS: Distance Formula with Solutions Available Online

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PART II: FINDING THE MIDPOINT OF A SEGMENT

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Vocabulary

midpoint of a segment - ___________________________________________________________________________________

point on a segment which divides the segment into two congruent segments

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In Coordinate Geometry, there are several ways to

determine the midpoint of a line segment.

Method 1:If the line segments are vertical or horizontal, you may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints.  

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Method 1: Horizontal or Vertical Lines

If the line segments are vertical or horizontal, you may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints. 

 

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To find the coordinates of the midpoint of a segment

when the lines are diagonal, we need to find the average (mean) of the coordinates of

the midpoint.

The Midpoint Formula: The midpoint of a segment endpoints (x1 , y1) and (x2 , y2) has coordinates                                                                   

The Midpoint Formula works for all line segments: 

vertical, horizontal or diagonal.

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Finding the MidpointFind the midpoint of

line segment AB. A (-3,4) B (2,1)

Check your answer here:

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Consider this “tricky” midpoint problem:

M is the midpoint of segment CD.  The coordinates M(-1,1) and C(1,-3) are given.  Find the coordinates of point D.

First, visualize the situation.  This will give you an idea of approximately where point D will be located.  When you find your answer, be sure it matches with your visualization of where the point should be

located.

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Solve algebraically:M(-1,1), C(1,-3) and D(x,y)Substitute into the Midpoint Formula:

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Solve for each variable separately:

(-3,5)

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Other Methods of Solution:Verbalizing the algebraic solution: Some students like to do these "tricky" problems by just examining the coordinates and asking themselves the following questions:"My midpoint's x-coordinate is -1.  What is -1 half of? (Answer -2)What do I add to my endpoint's x-coordinate of +1 to get -2? (Answer -3)This answer must be the x-coordinate of the other endpoint."These students are simply verbalizing the algebraic solution.  (They use the same process for the y-coordinate.)

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Final Checks for Understanding

1. Name two ways to find the midpoint of a given segment.

2. What method for finding the midpoint of a segment works for all lines…horizontal, vertical, and diagonal?

3. Explain how to find the coordinates of an endpoint when you are given an endpoint and the midpoint of a segment.

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Homework Assignment:

Page 46, text: 1-17 odd.*Extra Practice WS: Midpoint Formula with Solutions Available Online

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Solution Given: A(-3,4); B(2,1)

The midpoint will have coordinates:

Alert! Your answer may contain a

fraction.  Answers may be written in fractional or decimal form.

Answer:

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