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Wize High School Geometry Textbook (Common Core) > Foundations of Geometry

Midpoint of a Line Segment

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Midpoint of a Line Segment


There are so many things we can construct. Here is a useful one that will come in handy.

The Midpoint

The point that separates a line segment into two congruent parts is called the midpoint of the line segment.
Other objects can also be used to separate a line segment into two congruent parts such as a line, ray, or a plane. The object used to do this is called the segment bisector.

Example
Draw a diagram with two separate segments, labeled CD‾,\overline{CD}, and EF‾\overline{EF}.
In this diagram the segments should intersect each other at a point called GG.
Make sure that GG the midpoint of CD‾\overline{CD}, and that it is not the midpoint of EF‾\overline{EF}.

ANSWER:

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Constructing a segment bisector using a reflection

Our goal in this construction find the midpoint of a given line segment.

Using a compass and straightedge

Begin with a given line segment AB‾\overline{AB}.
  1. Fold your paper so that point AA and BB match up.
  2. Unfold the paper and mark the intersection of the crease and AB‾\overline{AB} as point CC.
From this we now have AC‾≅CB‾\overline{AC} \cong \overline{CB}


Using Technology


Begin with a given line segment AB‾\overline{AB}.
  1. Select the midpoint tool and click on point AA , then point BB.
  2. Label the new point as point CC.
From this we now have AC‾≅CB‾\overline{AC} \cong \overline{CB}
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Reasoning with the Midpoint



The diagram below represents your path to school, along with the various places you pass by.
Use it and the given information to answer the questions.


Starting at home, point AA, you begin your trip to school located at point EE.
When you reach point CC, you realize that you've completely forgotten to take your lunch. At this point you have two options:
  • Go back home, grab your lunch, and continue to school.
  • Go to school, and then at lunch time visit the store at point DD.

If point CC is the midpoint of AE‾\overline{AE}, and point DD is the midpoint of CE‾\overline{CE}, what choice will make you travel the most distance? How much more?

ANSWER: Going back home will be exactly twice as long as visiting the store.
To see why let's compare both trip options.
  • Going back home for lunch you will travel along the segment of AC‾\overline{AC} twice.
  • Visiting the store for lunch you will travel along the segment of DE‾\overline{DE} twice.
Since DD is the midpoint of CE‾\overline{CE}, the distance to travel to the store is the same as traveling CE‾\overline{CE}. Now since CC is the midpoint of AE‾\overline{AE}, then CE‾≅AC‾\overline{CE} \cong \overline{AC}. So the entire trip to the store and back is the length of AC‾\overline{AC}, which you will cover twice if you grab your lunch from home.
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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.