1. Find the distance of AB
Geometry

2. Geometry

3. Geometry

4. 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
Geometry

5. PART I: FINDING DISTANCE
Geometry

6. The Coordinate Plane The Coordinate Plane Origin  Quadrant II (-, +)
The coordinates of point T are ________.
(6,3)
(0,0)
Origin 
Quadrant III (-, -)
Quadrant IV (+, -)
The Coordinate Plane
Mrs. McConaughy
Geometry

7. 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 horizontal
or vertical, the length can be obtained
by counting.  
Mrs. McConaughy
Geometry

8. Method One 7 3 Unfortunately, this counting approach does NOT work for
EF which is a diagonal segment.
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 = ___) 7 3
Geometry

9. The distance, d, between two points, A(x1, y1) and B(x2, y2), is
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 for
all line segments: vertical, horizontal, and
diagonal.
Mrs. McConaughy
Geometry

10. ALERT! Order is important when using
Finding Distance
ALERT! Order is important when using
Distance Formula.
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.
(6,8)
(0,0)
(6,8)
(0,0)
Geometry

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

12. Applying the Distance Formula
Each morning H. I. Achiever takes the “bus line” from Oak to Symphony. How far is the bus ride from Oak to Symphony?
(2,4)Jackson
(__,__) North
(__,__) Central
(__,__) Symphony
(__,__) Cedar
(__,__) City Plaza
(__,__) Oak
Geometry

13. Geometry

14. Final Checks for Understanding
State the Distance Formula in words.
When should the Distance Formula be used when determining the distance between two given points?
Find the length of segment AB given A (-1,-2) and B (2,4).
Mrs. McConaughy
Geometry

15. Homework Assignment Page 46, text: 1-17 odd.
*Extra Practice WS: Distance Formula with Solutions Available Online
Geometry

16. PART II: FINDING THE
MIDPOINT OF A SEGMENT
Geometry

17. Vocabulary
midpoint of a segment - _______________ __________________________________ __________________________________
point on a segment which divides the segment into two congruent segments
Geometry

18. Geometry

19. 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.  
Mrs. McConaughy
Geometry

20. 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. 
Geometry

21. The Midpoint Formula works for all line segments:
vertical, horizontal or diagonal.
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                                                                   
Mrs. McConaughy
Geometry

22. Finding the Midpoint Find the midpoint of line segment AB. A (-3,4)
Check your answer here:
Geometry

23. 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.
Geometry

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

25. Solve for each variable separately:
(-3,5)
Mrs. McConaughy
Geometry

26. 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.)
Mrs. McConaughy
Geometry

27. Final Checks for Understanding
Name two ways to find the midpoint of a given segment.
What method for finding the midpoint of a segment works for all lines…horizontal, vertical, and diagonal?
Explain how to find the coordinates of an endpoint when you are given an endpoint and the midpoint of a segment.
Geometry

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

29. Solution Answer: 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:
Geometry
Click here to return to lesson.