Introduction for distinct point:
The distance between any two different points (x1, y1) and (x2, y2). The distance between two different points is basic concept in geometry. We now give an algebraic expression for the same. Let P1 (x1, y1) and P2(x2, y2) be two distinct points in the Cartesian plane and denote the distance between P1 and P2 by d(P1, P2) or by P1P2. Draw the line segment P1P2. There are three cases are following.
Cases for Distinct Point
Case (i):
The segment `bar (P_(1)P_(2))` is parallel to the x-axis. Then y1 = y2. Illustrate P1L and P2M, perpendicular in the direction of the y-axis. Then d(P1, P2) is equal to the distance between L and M. But L is (x1, 0) and M is (x2, 0). So the length LM = |x1 – x2|. Hence d(P1, P2) = |x1 – x2|.
Case (ii):
The segment `bar (P_(1)P_(2))` is parallel to the y-axis. Then x1 = x2. Illustrate P1L and P2M, perpendicular in the direction of the y-axis. Then d(P1, P2) is equal to the distance between L and M. But L is (0, y1) and M is (0, y2). So the length LM = |y1 – y2|. Hence d(P1, P2) = |y1 – y2|.
Case (iii):
The line segment `bar (P_(1)P_(2))` is neither parallel to the x-axis nor parallel to the y-axis. Draw a line through P1 parallel to x-axis and a line through P2 parallel to y-axis. Let these lines intersect at the point P3. Then P3 (x2, y1). The length of the line segment P1P3 is |x1-x2| and the length of the segment P3P2 is |y1-y2|. We observe that the triangle ΔP1P3P2 is a right triangle.
Formula for distinct point:
`sqrt((x_(2) - x_(1)^(2)) + (y_(2) - y_(1))^(2))`
Problems for Distinct Points:
Let us some problems of distinct points:
Problem 1:
Find the distance between the points A(10, 5) and B(4, 8).
Solution:
Let d is the distance between the two points A and B.
Formula for distinct point:
`sqrt((x_(2) - x_(1))^(2)) + (y_(2) - y_(1))^(2))`
` = sqrt((4 - 10^(2)) + (8 - 5)^(2))`
`= sqrt( ((-6)^(2)) + (3)^(2))`
`= sqrt (36 + 9)`
` =sqrt ( 45)`
` = 3sqrt ( 5)`
So, the dietance is `3sqrt(5)`
Problem 2:
Find the distance between the points A(7, 11) and B(20, 10).
Solution:
Let d is the distance between the two points A and B.
Formula for distinct point:
`sqrt((x_(2) - x_(1)^(2)) + (y_(2) - y_(1))^(2))`
`= sqrt((11 - 7^(2)) + (20 - 10)^(2))`
`= sqrt( ((4)^(2)) + (10)^(2))`
`= sqrt ( 160)`
` = 4sqrt ( 10)`
`These are problems of distinct points.`
The distance between any two different points (x1, y1) and (x2, y2). The distance between two different points is basic concept in geometry. We now give an algebraic expression for the same. Let P1 (x1, y1) and P2(x2, y2) be two distinct points in the Cartesian plane and denote the distance between P1 and P2 by d(P1, P2) or by P1P2. Draw the line segment P1P2. There are three cases are following.
Cases for Distinct Point
Case (i):
The segment `bar (P_(1)P_(2))` is parallel to the x-axis. Then y1 = y2. Illustrate P1L and P2M, perpendicular in the direction of the y-axis. Then d(P1, P2) is equal to the distance between L and M. But L is (x1, 0) and M is (x2, 0). So the length LM = |x1 – x2|. Hence d(P1, P2) = |x1 – x2|.
Case (ii):
The segment `bar (P_(1)P_(2))` is parallel to the y-axis. Then x1 = x2. Illustrate P1L and P2M, perpendicular in the direction of the y-axis. Then d(P1, P2) is equal to the distance between L and M. But L is (0, y1) and M is (0, y2). So the length LM = |y1 – y2|. Hence d(P1, P2) = |y1 – y2|.
Case (iii):
The line segment `bar (P_(1)P_(2))` is neither parallel to the x-axis nor parallel to the y-axis. Draw a line through P1 parallel to x-axis and a line through P2 parallel to y-axis. Let these lines intersect at the point P3. Then P3 (x2, y1). The length of the line segment P1P3 is |x1-x2| and the length of the segment P3P2 is |y1-y2|. We observe that the triangle ΔP1P3P2 is a right triangle.
Formula for distinct point:
`sqrt((x_(2) - x_(1)^(2)) + (y_(2) - y_(1))^(2))`
Problems for Distinct Points:
Let us some problems of distinct points:
Problem 1:
Find the distance between the points A(10, 5) and B(4, 8).
Solution:
Let d is the distance between the two points A and B.
Formula for distinct point:
`sqrt((x_(2) - x_(1))^(2)) + (y_(2) - y_(1))^(2))`
` = sqrt((4 - 10^(2)) + (8 - 5)^(2))`
`= sqrt( ((-6)^(2)) + (3)^(2))`
`= sqrt (36 + 9)`
` =sqrt ( 45)`
` = 3sqrt ( 5)`
So, the dietance is `3sqrt(5)`
Problem 2:
Find the distance between the points A(7, 11) and B(20, 10).
Solution:
Let d is the distance between the two points A and B.
Formula for distinct point:
`sqrt((x_(2) - x_(1)^(2)) + (y_(2) - y_(1))^(2))`
`= sqrt((11 - 7^(2)) + (20 - 10)^(2))`
`= sqrt( ((4)^(2)) + (10)^(2))`
`= sqrt ( 160)`
` = 4sqrt ( 10)`
`These are problems of distinct points.`
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