Sec' 1'3 Use Midpoint and Distance Formulas

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Sec. 1.3Use Midpoint and Distance Formulas

• The midpoint of a segment is the point that
  divides the segment into two congruent segments.

• A segment bisector is a point, ray, line, line
  segment, or plane that intersects the segment at
  its midpoint.
• A midpoint or a segment bisector bisects a
  segment.

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EXAMPLE 2
Use algebra with segment lengths
SOLUTION
VM MW
Write equation.
4x 1 3x 3
Substitute.
x 1 3
Subtract 3x from each side.
x 4
Add 1 to each side.
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EXAMPLE 2
Use algebra with segment lengths
VM 4x 1 4(4) 1 15
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for Examples 1 and 2
GUIDED PRACTICE
In Exercises 1 and 2, identify the segment
bisectorof PQ . Then find PQ.
1.
SOLUTION
Segment addition postulate.
Substitute
Add.
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for Examples 1 and 2
GUIDED PRACTICE
2.
SOLUTION
Point M is midpoint and line l bisects the line
PQ. Line l is the segment bisector of PQ.
So PM MQ. Now find PQ.
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for Examples 1 and 2
GUIDED PRACTICE
PM MQ
Write equation.
5x 7 11 2x
Substitute.
7x 18
Add 2x and 7 each side.
Divide each side by 7.
PQ 5x 7 11 2x 3x 4
Simplify.
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The Midpoint Formula
To find the coordinates of the midpoint of a
segment, we need to find the averages of the
x-coordinates and of the y-coordinates of the
endpoints.
If A(x1, y1) and B(x2, y2) are points in a
coordinate plane, then the midpoint M of AB has
coordinates
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EXAMPLE 3
Use the Midpoint Formula
a. FIND MIDPOINT The endpoints of RS are
R(1,3) and S(4, 2). Find the coordinates of the
midpoint M.
SOLUTION
a. FIND MIDPOINT Use the Midpoint Formula.
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EXAMPLE 3
Use the Midpoint Formula
SOLUTION
a. FIND MIDPOINT Use the Midpoint Formula.
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EXAMPLE 3
Use the Midpoint Formula
4 y 2
1 x 4
y 2
x 3
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for Example 3
GUIDED PRACTICE
SOLUTION
Use the midpoint formula.
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for Example 3
GUIDED PRACTICE
4. The midpoint of VW is M( 1, 2). One
endpoint is W(4, 4). Find the coordinates of
endpoint V.
SOLUTION
Let (x, y) be the coordinates of endpoint V. Use
the Midpoint Formula.
4 x 2
4 y 4
x 6
y 8
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If A(x1, y1) and B(x2, y2) are points in a
coordinate plane, then the distance between A and
B is
The distance formula is based on the Pythagorean
Theorem, which we will use in Ch 7.
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EXAMPLE 4
Standardized Test Practice
SOLUTION
Use the Distance Formula. You may find it helpful
to draw a diagram.
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EXAMPLE 4
Standardized Test Practice
Distance Formula
Substitute.
Subtract.
Evaluate powers.
Add.
Use a calculator to approximate the square root.
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for Example 4
GUIDED PRACTICE
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Homeworkp.19-214-8e, 12-26e, 32-40e, 44, 48