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Wize High School Grade 10 Math Textbook > Analytic Geometry

Midpoint of a Line Segment

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Midpoint of a Line Segment


When given a line segment, the midpoint is the point (x, y)\left(x,\ y\right) that is directly in between the endpoints of the line.


The midpoint formula is
M=(x1+x22, y1+y22)\Large\boxed{\displaystyle M=\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)}


Wize Tip
Always label your points carefully! Most mistakes happen because of unlabelled or mislabelled points!

You get to pick which endpoint is (x1, y1)(x_1,\ y_1) and which is (x2, y2)(x_2,\ y_2).

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Example: Midpoint of a Line Segment

Find the midpoint between (4,5)\left(4,5\right) and (−3, −6)\left(-3,\ -6\right).

First label the points

x1y1(4,5)\begin{array}{ccccc} &\colorTwo{x_1}&&\colorTwo {y_1}\\ (&\colorTwo 4&,&\colorTwo 5&) \end{array} and x2y2(−3,−6)\begin{array}{ccccc} &\colorThree{x_2}&&\colorThree {y_2}\\ (&\colorThree {-3}&,&\colorThree {-6}&) \end{array}

Let's calculate the midpoint for xx first:

x1+x22=4+(−3)2=12\displaystyle\frac{\colorTwo{x_1}+\colorThree{x_2}}{2}=\frac{\colorTwo4+\left(\colorThree{-3}\right)}{2}=\frac{1}{2}

The xx-coordinate of the midpoint is 12\dfrac{1}{2}.

Let's calculate the midpoint for yy:

y1+y22=5+(−6)2=−12\displaystyle\frac{\colorTwo{y_1}+\colorThree{y_2}}{2}=\frac{\colorTwo 5+\left(\colorThree {-6}\right)}{2}=\frac{-1}{2}

The yy-coordinate of the midpoint is −12-\dfrac{1}{2}

ANSWER: The midpoint is (12, −12)\boxed{\left(\dfrac{1}{2},~-\dfrac{1}{2}\right)}.

Practice: Midpoint of a Line Segment


Find the midpoint between (3,6)\left(3,6\right) and (−5, −10)\left(-5,\ -10\right).

Practice: Midpoint of a Line Segment

The center of a square has coordinates (−2, 5)(-2,~5) and the coordinates of the top left corner of the square is (−5, 8)(-5,~8). Find the coordinates of the other corners of this square.

Practice: Reflecting a Point Across a Line

The point (2,4)(2, 4) is reflected along the line y=−x−4y=-x-4. Find the coordinates of the point of reflection.