Download Measuring Segments

MEASURING AND
CONSTUCTING SEGMENTS
GEOMETRY (HOLT 1-2)
K. SANTOS
COORDINATE
A point corresponds to one and only one number on
a ruler. This number is called a coordinate.
A
1
2
A has coordinate 1
3
RULER POSTULATE 1-2-1
• The points of a line can be put into one-to-one
correspondence with the real numbers.
A
a
Where a and b are numbers
B
b
DISTANCE & LENGTH
Distance between any two points is the absolute
value of the difference of the coordinates.
A
B
a
b
AB = |a - b|or |b - a|
The distance between A and B is also called the
length of 𝐴𝐵
FINDING THE LENGTH OF A SEGMENT
Find AD, and DF.
A B C D E F G H I
-5 -4 -3 -2 -1 0 1 2 3
AD = −5 − (−2)
AD = −5 + 2
AD = −3
AD = 3
Remember all lengths are positive
DF = −2 − 0
DF = −2
DF = 2
DISTANCE BETWEEN POINTS
If G is at -12 and H is at 82, find the distance between
the points.
−12 − (82)
−94
94
So, the distance between G and H is 94 units.
What is the length of GH?
GH = 94 length and distance have the same
value
CONGRUENT SEGMENTS
• Two segments with the same length
• Symbol is
≅
• Pictures are marked with “tick” marks (same
number of marks on each congruent piece)
B
C
D
A
If AB = CD then 𝐴𝐵 ≅ 𝐶𝐷
Which is read: If the measure of AB equals the
measure of CD then AB is congruent to CD
EXAMPLE--CONGRUENT SEGMENTS
Find which two of the segments 𝑋𝑌, 𝑍𝑌, and 𝑍𝑊are
congruent.
X
Y
Z
W
-5
-1 2
6
XY = −5 − (−1)
XY = −5 + 1
XY = −4
XY = 4
ZY = 2 − (−1)
ZY = 2 + 1
ZY = 3
ZY = 3
So 𝑋𝑌 ≅ 𝑍𝑊 because XY = ZW
ZW = 2 − 6
ZW= −4
ZW = 4
ZW = 4
BETWEEN
In order to say that a point B is between two points A
and C, all three points must lie on the same line, and
AB + BC = AC.
A
B
C
B is between A and C
A
C
B
B is not between A and C
SEGMENT ADDITION POSTULATE 1-2-2
• If B is between A and C, then AB + BC = AC
• Notice A, B and C must be collinear
A
B
3
A
C
5
B
C
AB + BC = AC
3 + 5 = AC
8 = AC
ANOTHER SEGMENT ADDITION
POSTULATE EXAMPLE
If MN = 2x – 6, NO = x + 7 and MO = 25, find the value of x. Then find MN and
NO.
M
N
MN + NO = MO
2x – 6 + x + 7 = 25
3x + 1= 25
3x = 24
x=8
O
MN = 2x - 6
MN = 2(8) - 6
MN = 16-6
MN = 10
Check: MN + NO = MO remember MO = 25
10 + 15 = MO
25 = MO so it checks
NO = x + 7
NO = 8 + 7
NO = 15
MIDPOINT
• A point that divides the segment into two congruent
segments
R
If given S is the midpoint of 𝑅𝑇,
then 𝑅𝑆 ≅ 𝑆𝑇
S
T
MIDPOINT EXAMPLES
Given: Y is a midpoint of 𝐴𝐵
If XY = 6 find YZ and XZ.
YZ = 6
XZ = 2(6)=12
X
If XZ = 20 find XY and YZ
XY =
20
2
= 10
XZ = 10 (because XY = XZ)
Y
Z
ANOTHER MIDPOINT EXAMPLE
M is the midpoint of 𝑅𝑇. Find RM, MT and RT.
5x + 9
8x - 36
R
M
T
RM = TM
5x + 9 = 8x – 36
RM = 5x + 9
MT = 8x - 36
-3x +9 = -36
RM = 5(15) + 9 MT = 8(15) - 36
-3x = -45
RM = 75 + 9
MT = 120 - 36
x = 15
RM = 84
=
MT = 84
RT = RM + MT
RT = 84 + 84
RT = 168
SEGMENT BISECTOR
Segment Bisector—is any ray, segment, or line that
intersects a segment at its midpoint.
It divides the segment into two equal parts at its
midpoint.
D
C
A
B
𝐵𝐷 bisects 𝐴𝐶