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Midpoint: the point that divides the segment into two
congruent segments
Segment bisector: the point, ray, line, line segment or plane
that intersects the segments at its midpoint.
EXAMPLE 1
Find segment lengths
Skateboard
In the skateboard design, VW bisects XY at point T,
and XT = 39.9 cm. Find XY.
SOLUTION
Point T is the midpoint of XY .
So, XT = TY = 39.9 cm.
XY = XT + TY
Segment Addition Postulate
= 39.9 + 39.9 Substitute.
Add.
= 79.8 cm
EXAMPLE 2
Use algebra with segment lengths
ALGEBRA Point M is the midpoint
of VW . Find the length of VM .
SOLUTION
STEP 1
Write and solve an equation. Use the fact
that VM = MW.
VM = MW
4x – 1 = 3x + 3
x–1=3
x=4
Write equation.
Substitute.
Subtract 3x from each side.
Add 1 to each side.
EXAMPLE 2
STEP 2
Use algebra with segment lengths
Evaluate the expression
for VM when x = 4.
VM = 4x – 1 = 4(4) – 1 = 15
So, the length of VM is 15.
Check: Because VM = MW, the length of MW should be
15. If you evaluate the expression for MW, you should
find that MW = 15.
MW = 3x + 3 = 3(4) +3 = 15
GUIDED PRACTICE
for Examples 1 and 2
In Exercises 1 and 2, identify the segment bisector
of PQ . Then find PQ.
1.
ANSWER
3
MN; 3 4
GUIDED PRACTICE
for Examples 1 and 2
In Exercises 1 and 2, identify the segment bisector
of PQ . Then find PQ.
2.
ANSWER
5
line l ; 11
7
The Midpoint Formula
If A(X1, Y1) and B(X2, Y2) are points in a coordinate
plane, then the midpoint M of AB has coordinates
X1 + X2, Y1 + Y2
2
2
EXAMPLE Use the Midpoint Formula
3
a. FIND MIDPOINT The endpoints of RS are
R(1,–3) and S(4, 2). Find the coordinates of
the midpoint M.
EXAMPLE Use the Midpoint Formula
3
SOLUTION
a.
FIND MIDPOINT Use the Midpoint
Formula.
,– 3 + 2 = M 5,– 1
M1+
2
2 2
42
ANSWE
R
The coordinates of the midpoint M
are 5 ,– 1
2 2
EXAMPLE Use the Midpoint Formula
3
b. FIND ENDPOINT The midpoint of JK is
M(2, 1). One endpoint is J(1, 4). Find the
coordinates of endpoint K.
EXAMPLE Use the Midpoint Formula
3
SOLUTION
FIND ENDPOINT Let (x, y) be the
coordinates of endpoint K. Use
the Midpoint Formula.
STEP 1 Find x.
1+ x = 2
2
1+x=4
x=3
STEP 2 Find y.
4+ y 1
=
2
4+y=2
y=–
2
ANSWE The coordinates of endpoint K are (3, – 2
R
GUIDED PRACTICE for Example 3
3.
The endpoints of AB are A(1, 2) and B(7, 8).
Find the coordinates of the midpoint M.
ANSWE (4,5)
R
4. The midpoint of VW is M(– 1, – 2). One
endpoint is W(4, 4). Find the coordinates
of endpoint V.
ANSWE (– 6, – 8)
R
Homework:
P. 19: 1-29 odd
EXAMPLE 4 Standardized Test Practice
SOLUTION
Use the Distance Formula. You
may find it helpful to draw a
diagram.
EXAMPLE 4 Standardized Test Practice
2
2
RS = (x2 – x1 ) + (y
2 –1 y )
=
=
=
=
~
=
2
Distance Formula
2
[(4 – 2)] + [(–1) –3]Substitute.
2
2
(2) + (–4
)
4+1
6
2
0
4.47
Subtract.
Evaluate powers.
Add.
Use a calculator to approximate
the square root.
ANSWE The correct answer is C.
R
GUIDED PRACTICE for Example 4
5.
In Example 4, does it matter which ordered
1 (x , y ) and
pair you choose to substitute1 for
2 2
which ordered pair you choose to substitute
for (x , y )? Explain.
SAMPLE
ANSWER
No, when
squaring the differences in the
coordinates, you get the same answer as long
as you choose the x and y values from the
same point.
GUIDED PRACTICE for Example 4
6.
What is the approximate length of AB , with
endpoints A(–3, 2) and B(1, –4)?
6.1
7.2 units
8.5 units
10.0 units
units
ANSWE
R
B
Homework:
p. 20: 31-49 odd, plus 48