Transcript 2. - SharpSchool

1-2 Measuring and Constructing Segments
Warm Up
Solve each equation.
1. 2x – 6 = 7x – 31 2
2. 1/4 x – 6 = 220
904
3. 3(5 – 2x) = -3x
5
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Objectives
Use length and midpoint of a segment.
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Vocabulary
coordinate
midpoint
distance
bisect
length
segment bisector
construction
between
congruent segments
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
The distance between any two points is the
absolute value of the difference of the coordinates.
The distance between A and B is also called the
length of AB, or AB.
A
a
Holt McDougal Geometry
B
b
AB = |a – b| or |b - a|
1-2 Measuring and Constructing Segments
Example 1: Finding the Length of a Segment
Find each length.
A. BC
B. AC
BC = |1 – 3|
AC = |–2 – 3|
= |1 – 3|
= |– 5|
=2
=5
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Check It Out! Example 1
Find each length.
a. XY
Holt McDougal Geometry
b. XZ
1-2 Measuring and Constructing Segments
In order for you to say that a point B is
between two points A and C, all three
points must lie on the same line –
collinear.
•
AB + BC = AC
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Example 3A: Using the Segment Addition Postulate
G is between F and H, FG = 6, and FH = 11.
Find GH.
Hint: First draw the diagram.
FH = FG + GH
11 = 6 + GH
– 6 –6
5 = GH
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Example 3a
Y is between X and Z, XZ = 3, and XY =
Find YZ.
XZ = XY + YZ
Holt McDougal Geometry
.
1-2 Measuring and Constructing Segments
TRY THIS…
M is between N and O.
Find NO.
NM + MO = NO
17 + (3x – 5) = 5x + 2
3x + 12 = 5x + 2
–2
–2
3x + 10 = 5x
–3x
–3x
10 = 2x
2
2
5=x
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Check Your Work!!!!!!!
M is between N and O.
Find NO.
NO = 5x + 2
= 5(5) + 2
= 27
Holt McDougal Geometry
Substitute 5 for x.
Simplify.
1-2 Measuring and Constructing Segments
Check It Out! Example 3b
E is between D and F. Find DF.
DE + EF = DF
(3x – 1) + 13 = 6x
3x + 12 = 6x
– 3x
– 3x
12 = 3x
12 3x
=
3
3
4=x
Holt McDougal Geometry
Substitute the given values
1-2 Measuring and Constructing Segments
Check Your Work!
E is between D and F. Find DF.
DF = 6x
= 6(4)
Substitute 4 for x.
= 24
Simplify.
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Congruent segments are segments that have
the same length.
In the diagram, PQ = RS, so you can write
PQ  RS.
“Segment PQ is congruent to segment RS.”
Tick marks are used in a figure to show
congruent segments.
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
The midpoint (middle point) of AB is the
point that bisects (divides), the segment
into two congruent segments.
A
M
B
If M is the midpoint of AB, then AM = MB.
 So if AB = 6, then AM = 3 and MB = 3
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Example 5: Using Midpoints to Find Lengths
D is the midpoint of EF, ED = 4x + 6, and
DF = 7x – 9. Find ED, DF, and EF.
E
4x + 6
D
7x – 9
Step 1 Solve for x.
ED = DF
4x + 6 = 7x – 9
–4x
–4x
6 = 3x – 9
+9
+9
15 = 3x
x=5
Holt McDougal Geometry
F
1-2 Measuring and Constructing Segments
Always Check Your Work!!!!
D is the midpoint of EF, ED = 4x + 6, and
DF = 7x – 9. Find ED, DF, and EF.
E
4x + 6
D
7x – 9
F
Step 2 Find ED, DF, and EF.
DF = 7x – 9
ED = 4x + 6
= 7(5) – 9
= 4(5) + 6
= 26
= 26
Holt McDougal Geometry
EF = ED + DF
= 26 + 26
= 52
1-2 Measuring and Constructing Segments
DO NOW
S is the midpoint of RT, RS = –2x, and
ST = –3x – 2. Find RS, ST, and RT.
R
–2x
S
–3x – 2
T
Step 1 Solve for x.
S is the mdpt. of RT.
RS = ST
–2x = –3x – 2 Substitute –2x for RS and –3x – 2 for ST.
+3x +3x
x = –2
Holt McDougal Geometry
Can x be negative?
1-2 Measuring and Constructing Segments
Are you checking your work?????
S is the midpoint of RT, RS = –2x, and
ST = –3x – 2. Find RS, ST, and RT.
R
–2x
S
–3x – 2
T
Step 2 Find RS, ST, and RT.
RS = –2x
= –2(–2)
=4
Holt McDougal Geometry
ST = –3x – 2
= –3(–2) – 2
=4
RT = RS +
ST = 4 + 4
=8
1-2 Measuring and Constructing Segments
Lesson Quiz: Part I
1. M is between N and O. MO = 15, and MN =
7.6. Find NO.
22.6
2. S is the midpoint of TV, TS = 4x – 7, and
SV = 5x – 15. Find TS, SV, and TV.
25, 25, 50
3. LH bisects GK at M. GM = 2x +
6, and GK = 24. Find x.
3
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Round Table
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Independent Practice
P. 12 #6 – 12
P. 19 # 5, 11 – 13
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Do Now
Quick Write
1. Can a line have a midpoint or bisector?
Explain?
2. What is the difference between a point on a
line, and the midpoint? How does this
difference affect how you set up an equation
for each type of problem?
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Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Objective
• SWBAT calculate the length and midpoint of a
segment in a coordinate plane.
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Line Segments in a coordinate plane
• Vertices
• Midpoint
• Endpoints
• Length
Could you derive a formula to find the midpoint of
this line segment?
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Midpoint in a Coordinate Plane
The midpoint M of AB with endpoints
A( X1, Y1) and B(X2 , Y2) is found by:
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Example 1: Finding the Coordinates of a Midpoint
Find the coordinates of the midpoint of PQ
with endpoints P(–8, 3) and Q(–2, 7).
= (–5, 5)
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Check It Out! Example 1
Find the coordinates of the midpoint of EF
with endpoints E(–2, 3) and F(5, –3).
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
EXTENSION: Finding the Coordinates of an Endpoint
M is the midpoint of XY. X has coordinates
(2, 7) and M has coordinates (6, 1). Find
the coordinates of Y.
Step 1 Let the coordinates of Y equal (x, y).
Step 2 Use the Midpoint Formula:
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Example 2 Continued
Step 3 Find the x-coordinate.
12 = 2 + x
– –2
2
10 = x
The coordinates of Y are (10, –5).
Holt McDougal Geometry
2=7+y
– –7
7 –5 = y
1-2 Measuring and Constructing Segments
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Example 3: Using the Distance Formula
Find FG and JK.
Then determine whether FG  JK.
Step 1 Find the
coordinates of each
point.
F(1, 2), G(5, 5), J(–4,
0), K(–1, –3)
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Check It Out! Example 3
Find EF and GH. Then determine if EF  GH.
Step 1 Find
the
coordinates
of each
point.
E(–2, 1),
F(–5, 5),
G(–1, –2),
H(3, 1)
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Check It Out! Example 3 Continued
Step 2 Use the Distance Formula.
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Find the perimeter
of Triangle KLM
K (-2, 5)
L (1, 1)
M (-3, 1)
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Independent Practice
P. 19
#17 – 22
Challenge
P.20 #26 & 27
Holt McDougal Geometry
1-2 Measuring and Constructing Segments
Exit Ticket
Find the midpoint of AB when
A(6, -2) and B(8, -5)
Challenge Exit Ticket
Given line segment AB where A(5, 4)
and midpoint M(3, 3). What are the
coordinates of B?
Holt McDougal Geometry