1.4: Midpoints and Bisectors
Learning Objectives
 Identify the midpoint of line segments.
 Identify the bisector of a line segment.
 Understand and the Angle Bisector Postulate.
Review Queue
Answer the following questions.

m∠ROT=165∘ , findm∠POT  Find
x .  Use the Angle Addition Postulate to write an equation for the angles in #1.
Know What? The building to the right is the TransamericaBuilding in San Francisco. This building was completed in 1972 and, at that time was one of the tallest buildings in the world. It is a pyramid with two “wings” on either side, to accommodate elevators. Because San Francisco has problems with earthquakes, there are regulations on how a building can be designed. In order to make this building as tall as it is and still abide by the building codes, the designer used this pyramid shape.
It is very important in designing buildings that the angles and parts of the building are equal. What components of this building look equal? Analyze angles, windows, and the sides of the building.
Congruence
You could argue that another word for equal is congruent. However, the two differ slightly.
Congruent: When two geometric figures have the same shape and size.
We label congruence with a
Equal  Congruent 



used with measurement  used with figures 




Midpoints
Midpoint: A point on a line segment that divides it into two congruent segments.
Because
Midpoint Postulate: Any line segment will have exactly one midpoint.
This might seem selfexplanatory. However, be careful, this postulate is referring to the midpoint, not the lines that pass through the midpoint, which is infinitely many.
Example 1: Is
Solution: No, it is not because
Midpoint Formula
When points are plotted in the coordinate plane, you can use slope to find the midpoint between then. We will generate a formula here.
Here are two points, (5, 6) and (3, 4). Draw a line between the two points and determine the vertical distance and the horizontal distance.
So, it follows that the midpoint is down and over half of each distance. The midpoint would then be down 2 (or 2) from (5, 6) and over positively 4. If we do that we find that the midpoint is (1, 4).
Let’s create a formula from this. If the two endpoints are (5, 6) and (3, 4), then the midpoint is (1, 4). 1 is halfway between 5 and 3 and 4 is halfway between 6 and 2. Therefore, the formula for the midpoint is the average of the
Midpoint Formula: For two points,
Example 2: Find the midpoint between (9, 2) and (5, 14).
Solution: Plug the points into the formula.
Example 3: If
Solution: Plug what you know into the midpoint formula.
Another way to find the other endpoint is to find the difference between
Segment Bisectors
Segment Bisector: A line, segment, or ray that passes through a midpoint of another segment.
A bisector cuts a line segment into two congruent parts.
Example 4: Use a ruler to draw a bisector of the segment below.
Solution: The first step in identifying a bisector is finding the midpoint. Measure the line segment and it is 4 cm long. To find the midpoint, divide 4 by 2.
So, the midpoint will be 2 cm from either endpoint, or halfway between. Measure 2 cm from one endpoint and draw the midpoint.
To finish, draw a line that passes through the midpoint. It doesn’t matter how the line intersects
A specific type of segment bisector is called a perpendicular bisector.
Perpendicular Bisector: A line, ray or segment that passes through the midpoint of another segment and intersects the segment at a right angle.
Perpendicular Bisector Postulate: For every line segment, there is one perpendicular bisector that passes through the midpoint.
There are infinitely many bisectors, but only one perpendicular bisector for any segment.
Example 5: Which line is the perpendicular bisector of
Solution: The perpendicular bisector must bisect
Example 6: Algebra Connection Find
Solution: The line shown is the perpendicular bisector. So,
And,
Investigation 13: Constructing a Perpendicular Bisector
 Draw a line that is at least 6 cm long, about halfway down your page.
 Place the pointer of the compass at an endpoint. Open the compass to be greater than half of the segment. Make arc marks above and below the segment. Repeat on the other endpoint. Make sure the arc marks intersect.
 Use your straight edge to draw a line connecting the arc intersections.
This constructed line bisects the line you drew in #1 and intersects it at
Congruent Angles
Example 7: Algebra Connection What is the measure of each angle?
Solution: From the picture, we see that the angles are congruent, so the given measures are equal.
To find the measure of
Because
Angle Bisectors
Angle Bisector: A ray that divides an angle into two congruent angles, each having a measure exactly half of the original angle.
Angle Bisector Postulate: Every angle has exactly one angle bisector.
Example 8: Let’s take a look at Review Queue #1 again. Is
Solution: Yes,
Investigation 14: Constructing an Angle Bisector
 Draw an angle on your paper. Make sure one side is horizontal.
 Place the pointer on the vertex. Draw an arc that intersects both sides.
 Move the pointer to the arc intersection with the horizontal side. Make a second arc mark on the interior of the angle. Repeat on the other side. Make sure they intersect.
 Connect the arc intersections from #3 with the vertex of the angle.
To see an animation of this construction, view http://www.mathsisfun.com/geometry/constructanglebisect.html.
Know What? Revisited The image to the right is an outline of the Transamerica Building from earlier in the lesson. From this outline, we can see the following parts are congruent:
As well at these components, there are certain windows that are congruent and all four triangular sides of the building are congruent to each other.
Review Questions
 Copy the figure below and label it with the following information:
For 29, find the lengths, given:

AB 
GA 
ED 
HE 
m∠HDC 
FA 
GD 
m∠FED  How many copies of triangle
AHB can fit inside rectangleFECA without overlapping?
For 1118, use the following picture to answer the questions.
 What is the angle bisector of
∠TPR ? 
P is the midpoint of what two segments?  What is
m∠QPR ?  What is
m∠TPS ?  How does
VS¯¯¯¯¯¯¯ relate toQT¯¯¯¯¯¯¯¯ ?  How does
QT¯¯¯¯¯¯¯¯ relate toVS¯¯¯¯¯¯¯ ?  Is
PU¯¯¯¯¯¯¯¯ a bisector? If so, of what?  What is
m∠QPV ?
Algebra Connection For 1924, use algebra to determine the value of variable(s) in each problem.

Construction Using your protractor, draw an angle that is
110∘ . Then, use your compass to construct the angle bisector. What is the measure of each angle? 
Construction Using your protractor, draw an angle that is
75∘ . Then, use your compass to construct the angle bisector. What is the measure of each angle?  Construction Using your ruler, draw a line segment that is 7 cm long. Then use your compass to construct the perpendicular bisector, What is the measure of each segment?
 Construction Using your ruler, draw a line segment that is 4 in long. Then use your compass to construct the perpendicular bisector, What is the measure of each segment?

Construction Draw a straight angle
(180∘) . Then, use your compass to construct the angle bisector. What kind of angle did you just construct?
For questions 3033, find the midpoint between each pair of points.
 (2, 3) and (8, 7)
 (9, 1) and (6, 11)
 (4, 10) and (14, 0)
 (0, 5) and (9, 9)
Given the midpoint

A(−1,2) andM(3,6) 
B(−10,−7) andM(−2,1) 
Error Analysis Erica is looking at a geometric figure and trying to determine which parts are congruent. She wrote
AB¯¯¯¯¯¯¯¯=CD¯¯¯¯¯¯¯¯ . Is this correct? Why or why not? 
Challenge Use the Midpoint Formula to solve for the
x− value of the midpoint and they− value of the midpoint. Then, use this formula to solve #34. Do you get the same answer? 
Construction Challenge Use construction tools and the constructions you have learned in this section to construct a
45∘ angle.  Construction Challenge Use construction tools and the constructions you have learned in this section to construct two 2 in segments that bisect each other. Now connect all four endpoints with segments. What figure have you constructed?
 Describe an example of how the concept of midpoint (or the midpoint formula) could be used in the real world.
Review Queue Answers
 See Example 6

2x−5=33 2x=38 x=19 
m∠ROT=m∠ROS+m∠SOP+m∠POT
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