Midpoint and Distance in the Coordinate Plane
1-6
Midpoint and Distance in the Coordinate Plane Why learn this? You can use a coordinate plane to help you calculate distances. (See Example 5.)
Objectives Develop and apply the formula for midpoint. Use the Distance Formula and the Pythagorean Theorem to find the distance between two points. Vocabulary coordinate plane leg hypotenuse
Major League baseball fields are laid out according to strict guidelines. Once you know the dimensions of a field, you can use a coordinate plane to find the distance between two of the bases. A coordinate plane is a plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis). The location, or coordinates, of a point are given by an ordered pair (x, y). You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Midpoint Formula _ The midpoint M of AB with endpoints A(x 1, y 1) and B(x 2, y 2) is found by
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Preparation for 17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. Also covered: 15.0
EXAMPLE
To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane.
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x1 + x2 _ y + y2 , 1 . M _ 2 2
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Finding the Coordinates of a Midpoint
−− Find the coordinates of the midpoint of CD with endpoints C(-2, -1) and D(4, 2). y + y2 x1 + x2 _ M _ , 1 2 2
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-2 + 4 _ -1 + 2 2, _ 1 _ , = _ 2 2 2 2
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_ 1. Find the coordinates of the midpoint of EF with endpoints E(-2, 3) and F(5, -3). 1- 6 Midpoint and Distance in the Coordinate Plane
43
Midpoint and Distance in the Coordinate Plane Why learn this? You can use a coordinate plane to help you calculate distances. (See Example 5.)
Objectives Develop and apply the formula for midpoint. Use the Distance Formula and the Pythagorean Theorem to find the distance between two points. Vocabulary coordinate plane leg hypotenuse
Major League baseball fields are laid out according to strict guidelines. Once you know the dimensions of a field, you can use a coordinate plane to find the distance between two of the bases. A coordinate plane is a plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis). The location, or coordinates, of a point are given by an ordered pair (x, y). You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Midpoint Formula _ The midpoint M of AB with endpoints A(x 1, y 1) and B(x 2, y 2) is found by
(
Preparation for 17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. Also covered: 15.0
EXAMPLE
To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane.
)
ÛiÀ>}iÊvÊ Þ£Ê>`ÊÞÓ
ÝÓ]ÊÞÓ® Ý Ê ÊÝ Þ Ê ÊÞ Ê ]ÊÊÊÊpÊ Ê ÊÊ ÊÊpÊ Ó Ó ® £
Þ£ ä
1
ÞÓ
x1 + x2 _ y + y2 , 1 . M _ 2 2
California Standards
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Finding the Coordinates of a Midpoint
−− Find the coordinates of the midpoint of CD with endpoints C(-2, -1) and D(4, 2). y + y2 x1 + x2 _ M _ , 1 2 2
(
)
( ) 1 = (1, _ 2)
-2 + 4 _ -1 + 2 2, _ 1 _ , = _ 2 2 2 2
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_ 1. Find the coordinates of the midpoint of EF with endpoints E(-2, 3) and F(5, -3). 1- 6 Midpoint and Distance in the Coordinate Plane
43
