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The Distance Formula, Midpoint Formula, Defined Segment, Simplifying, Two Points, Pair of Points, line Segment has Coordinates, Coordinates of the Midpoint, Segment Joining are the key points of this lecture.
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Objectives: 1. Use the Distance Formula
Given any two points ( x 1 , y 1 ), ( x 2 , y 2 ), the length of the defined segment d is found using the following formula:
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Example 1 : Find the distance between the two points: (โ5, 8), (0, โ4)
Solution : Use the distance formula to find the distance between the two points.
Replace x 2 with 0, x 1 with โ5, y 2 with โ4 and y 1 with 8.
Simplifying:
Find the square root of 169. = 13 The distance between the pair of points is 13 units.
Example 2 : Find the distance between the two points:
Solution : Use the distance formula to find the distance between the two points.
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Given any two points ( x 1 , y 1 ), ( x 2 , y 2 ), the midpoint of the defined segment d is found using the following formula(s):
Example 3: Find the midpoint of the line segment connecting (6, 5) and (12, 9). Solution : The midpoint of a line segment has coordinates,
Replace x 1 with 6, x 2 with 12, y 1 with 5 and y 2 with 9.
Hence, the coordinates of the midpoint are (9, 7).
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Example 4: Find the midpoint of the line segment connecting (0.25, 4) and (โ0.9, โ2). Solution: The midpoint of a line segment has coordinates,
Replace x 1 with 0.25, x 2 with โ0.9, y 1 with 4 and y 2 with โ2.
Hence, the coordinates of the midpoint are (โ0.325, 1).
Example 5 : Determine the distance between the two points and then determine the midpoint of the line segment joining the two points (0, 0), (6, 8). Solution : The distance d between the points ( x 1 , y 1 ) and ( x 2 , y 2 ) is:
Substituting 0 for x 1 , 6 for x 2 and 0 for y 1 , 8 for y 2 in the Distance Formula:
d = 10 The distance between the points is 10. The midpoint of the line segment joining A ( x 1 , y 1 ) and B ( x 2 , y 2 ) is:
Substitute 0 for x 1 , 6 for x 2 and 0 for y 1 , 8 for y 2 in the Midpoint Formula:
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