1.8: Congruent Segments, Midpoints, and Bisectors
Learning Objectives
- Define midpoints, congruent, and bisectors.
- Use the Segment Addition Postulate to find lengths of segments.
- Use midpoints to find the lengths of segments.
Congruent Line Segments
One of the most important words in geometry is congruent. This term refers to geometric objects that have exactly the same size and shape. Two segments are congruent if they have the same length.
- Congruent objects have the _______________________ size and shape.
Notation Notes:
1. When two things are congruent we use the symbol \begin{align*}\cong\end{align*}.
For example if \begin{align*}\overline{AB}\end{align*} is congruent to \begin{align*}\overline{CD}\end{align*}, then we would write \begin{align*}\overline{AB} \cong \overline{CD}\end{align*}.
2. When we draw congruent segments, we use tic marks to show that two segments are congruent.
3. If there are multiple pairs of congruent segments (which are not congruent to each other) in the same picture, use two tic marks for the second set of congruent segments, three for the third set, and so on. See the two following illustrations:
\begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{CD}\end{align*} are congruent line segments.
\begin{align*}\overline{ED}\end{align*} and \begin{align*}\overline{FG}\end{align*} are one set of congruent line segments, and \begin{align*}\overline{DG}\end{align*} and \begin{align*}\overline{EF}\end{align*} are another set of congruent line segments.
- The geometric symbol for congruent is ____________.
The length of segment \begin{align*}\overline{AB}\end{align*} can be written in two ways: \begin{align*}m\overline{AB}\end{align*} or simply \begin{align*}AB\end{align*}. This might be a little confusing at first, but it will make sense the more you use this notation.
Let’s say we used a ruler and measured \begin{align*}\overline{AB}\end{align*} and we saw that it had a length of 5 cm. Then we could write \begin{align*}m\overline{AB} = 5 \ cm\end{align*}, or \begin{align*}AB = 5 \ cm\end{align*}.
If we know that \begin{align*}\overline{AB} \cong \overline{CD}\end{align*}, then we can write \begin{align*}m\overline{AB} = m\overline{CD}\end{align*} or simply \begin{align*}AB = CD\end{align*}.
Reading Check:
Line segment \begin{align*}\overline{CD}\end{align*} has the same length as line segment \begin{align*}\overline{RQ}\end{align*}.
How could you write this statement three different ways?
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Segment Midpoints
Now that you understand congruent segments, there are a number of new terms and types of figures you can explore.
A segment midpoint is a point on a line segment that divides the segment into two congruent segments. So, each segment between the midpoint and an endpoint will have the same length.
- A segment midpoint splits a segment into two _________________________ parts.
In the diagram below,
point \begin{align*}B\end{align*} is the midpoint of segment \begin{align*}\overline{AC}\end{align*} since \begin{align*}\overline{AB}\end{align*} is congruent to \begin{align*}\overline{BC}\end{align*}:
There is even a special postulate dedicated to midpoints.
Segment Midpoint Postulate
Any line segment will have exactly one midpoint—no more, and no less.
Reading Check:
1. Why can there only be one midpoint?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. How would you explain your answer above to a person who thought there could be three midpoints on one line segment?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Because a segment midpoint divides the segment into two congruent segments, you can use the midpoint to find the measure of a line segment.
In the picture below, point \begin{align*}M\end{align*} is the midpoint of segment \begin{align*}\overline{LN}\end{align*}.
If \begin{align*}\overline{LM} = 4 \ cm\end{align*}, then \begin{align*}\overline{MN}\end{align*} must also be equal to 4 cm, because the midpoint divides the segment into two congruent segments.
What would be the length of the entire segment \begin{align*}\overline{LN}\end{align*}? The entire segment is equal to 8 cm, because each half is equal to 4 cm, and \begin{align*}4 \ cm + 4 \ cm = 8 \ cm\end{align*}.
If \begin{align*}\overline{LM} =\end{align*} ______________ and \begin{align*}\overline{MN} =\end{align*} ______________ then \begin{align*}\overline{LN} =\end{align*} ______________.
Segment Bisectors
Now that you know how to find midpoints of line segments, you can explore segment bisectors.
A bisector is a line, segment, or ray that passes through a midpoint of another segment.
You probably know that the prefix “bi-” means two (think about the two wheels of a bicycle). So, a bisector cuts a line segment into two congruent parts.
- A bisector cuts a segment into two ___________________________ parts.
- A bisector passes through a line segment’s ___________________________.
Segment Addition Postulate
The measure of any line segment can be found by adding the measures of the smaller segments that comprise it.
That may seem like a lot of confusing words, but the logic is quite simple:
In the diagram below, if you add the lengths of \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{BC}\end{align*}, you will have found the length of \begin{align*}\overline{AC}\end{align*}. In geometric symbols: \begin{align*}AB + BC = BC\end{align*}.
- You can _________________ together all parts of a line segment to get the length of the entire segment.
For example, in the picture below, \begin{align*}m\overline{DE} = 2 \ cm\end{align*} and \begin{align*}m\overline{EF} = 6 \ cm\end{align*}. What is the total length of line segment \begin{align*}\overline{DF}\end{align*}?
Because of the Segment __________________________ Postulate, you can simply add the two parts of the segment together to get the total length of the line segment.
The length of line segment \begin{align*}\overline{DF}\end{align*} is 8 cm, because \begin{align*}2 \ cm + 6 \ cm = 8 \ cm\end{align*}.
Is \begin{align*}E\end{align*} the midpoint of line segment \begin{align*}\overline{DF}\end{align*}?
It is not the midpoint of \begin{align*}\overline{DF}\end{align*} because it does not divide the line segment into two congruent parts. \begin{align*}\overline{DE}\end{align*} is shorter than \begin{align*}\overline{EF}\end{align*}, so \begin{align*}\overline{DE}\end{align*} and \begin{align*}\overline{EF}\end{align*} are not congruent.
You can also use the Segment Addition Postulate to find missing measures of line segments within a larger line segment.
In the example below, \begin{align*}m\overline{GH} = 3 \ in\end{align*}. and \begin{align*}m\overline{GI} = 10 \ in\end{align*}. You can write an equation to help you find the length of \begin{align*}\overline{HI}\end{align*}:
You can subtract \begin{align*}GH\end{align*} from \begin{align*}GI\end{align*}. Since \begin{align*}GI = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*} and \begin{align*}GH = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}, GI - GH = 10 - 3 = 7\end{align*}
So the length of \begin{align*}\overline{HI} = 7\end{align*} inches. You will also notice that \begin{align*}3 \ in. + 7 \ in. = 10 \ in.\end{align*}
Reading Check:
1. Is \begin{align*}C\end{align*} the midpoint of line segment \begin{align*}\overline{XY}\end{align*} below? _____________________
Why or why not? Explain.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. Use the Segment Addition Postulate to find the length of segment \begin{align*}\overline{XY}\end{align*} in the figure above.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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