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Midpoint. Coordinates of midpoint

Definition.
Midpoint is a point that is located on the segment and is equidistant from the segment endpoints.
Midpoint

In geometric problems can often be faced with the need to find the midpoint of segment that defined by coordinates of its endpoints, for example in search problems median, midline, etc.

Each coordinate of the midpoint is equal to half the sum of the corresponding coordinates of the endpoints.


Midpoint formulas:

  • The formula for calculating the coordinates of midpoint of a line segment with endpoints A(xaya) and B(xbyb) on plane:

    xc xa + xb        yc ya + yb
    22

  • The formula for calculating the coordinates of midpoint of a line segment with endpoints A(xayaza) and B(xbybzb) in space:

    xc xa + xb      yc ya + yb      zc za + zb
    222

Examples of tasks with midpoint

Examples of tasks with coordinates of midpoint of line segment on the plane

Example 1.
Find the midpoint, point C, which is between the two endpoints A(-1, 3) and B(6, 5).

Solution.

xc
xa
+
xb
 =  -1 + 6  =  5  = 2.5
222
yc
ya
+
yb
 =  3 + 5  =  8  = 4
222

Answer: С(2.5, 4).

Example 2.
Find the Coordinates of endpoint, point B, when midpoint is C(1; 5) and other endpoint A(-1, 3).

Solution.

xc
xa
+
xb
  =>  
xb
 = 2
xc
 - 
xa
 = 2·1 - (-1) = 2 + 1 = 3
2
yc
ya
+
yb
  =>  
yb
 = 2
yc
 - 
ya
 = 2·5 - 3 = 10 - 3 = 7
2

Answer: B(3, 7).


Examples of tasks with coordinates of midpoint of line segment on the space

Example 3.
Find the midpoint, point C, which is between the two endpoints A(-1, 3, 1) and B(6, 5, -3).

Solution.

xc
xa
+
xb
 =  -1 + 6  =  5  = 2.5
222
yc
ya
+
yb
 =  3 + 5  =  8  = 4
222
zc
za
+
zb
 =  1 + (-3)  =  -2  = -1
222

Answer: С(2.5, 4, -1).

Example 4.
Find the Coordinates of endpoint, point B, when midpoint is C(1, 5, 2) and other endpoint A(-1, 3, 10).

Solution.

xc
xa
+
xb
  =>  
xb
 = 2
xc
 - 
xa
 = 2·1 - (-1) = 2 + 1 = 3
2
yc
ya
+
yb
  =>  
yb
 = 2
yc
 - 
ya
 = 2·5 - 3 = 10 - 3 = 7
2
zc
za
+
zb
  =>  
zb
 = 2
zc
 - 
za
 = 2·2 - 10 = 4 - 10 = -6
2

Answer: B(3, 7, -6).

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