1.1 Segment Length and Midpoints

Transcription

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Name
1.1
Class
Date
Segment Length and Midpoints
Essential Question: How do you draw a segment and measure its length?
Resource
Locker
Explore
Exploring Basic Geometric Terms
In geometry, some of the names of figures and other terms will already be
familiar from everyday life. For example, a ray like a beam of light from a
spotlight is both a familiar word and a geometric figure with a mathematical
definition.
The most basic figures in geometry are undefined terms, which cannot be
defined using other figures. The terms point, line, and plane are undefined terms.
Although they do not have formal definitions, they can be described as shown in
the table.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Marco
Vacca/age fotostock
Undefined Terms
Term
Geometric Figure
A point is a specific location. It has
no dimension and is represented by
a dot.
P
A line is a connected straight path.
It has no thickness and it continues
forever in both directions.
A plane is a flat surface. It has no
thickness and it extends forever in
all directions.
B
A
point P
ℓ
X
line ℓ, line AB, line BA,
‹ › −
‹ ›
−
AB, or BA
Z

Ways to Name the Figure
Y
plane  or plane XYZ
In geometry, the word between is another undefined term, but its meaning is understood
from its use in everyday language. You can use undefined terms as building blocks
to write definitions for defined terms, as shown in the table.
Defined Terms
Term
Geometric Figure
A line segment (or segment) is a
portion of a line consisting of two
points (called endpoints) and all
points between them.
A ray is a portion of a line that
starts at a point (the endpoint) and
continues forever in one direction.
Module 1
GE_MNLESE385795_U1M01L1.indd 5
C
D
P
Q
5
Ways to Name the Figure
segment
_
_CD, segment DC,
CD, or DC
→
‾
ray PQ or PQ
Lesson 1
3/20/14 5:03 PM
DO NOT
Correcti
You can use points to sketch lines, segments, rays, and planes.
A
B
Draw two points J and K. Then draw a line
through them. (Remember that a line
shows arrows at both ends.)
Draw two points J and K again. This time, draw
the line segment with endpoints J and K.
J
J
K
K
C
D
Draw a point K again and draw a ray from
endpoint K. Plot a point J along the ray.
Draw three points J, K, and M so that they are
not all on the same line. Then draw the plane
that contains the three points. (You might also
put a script letter such as  on your plane.)
J
J
M
K
E
K

Give a name for each of the figures you drew. Then use a circle to choose whether the type
of figure is an undefined term or a defined term.
Point
Line
Segment
Plane
undefined term/defined term
‹ ›
‹ ›
−
−
JK (or KJ )
_
_
JK or KJ
→
‾
KJ
plane JKM (or plane )
Reflect
1.
2.
→
→
‾ be the same ray as KJ
‾ ? Why or why not?
In Step C, would JK
No. The rays would have different endpoints and continue in opposite directions.
In Step D, when you name a plane using 3 letters, does the order of the letters matter?
No. Using 3 letters, the plane in Step D can be named plane JKM, plane JMK, plane KJM,
© Houghton Mifflin Harcourt Publishing Company
Ray
points J, K, and M
plane KMJ, plane MJK, or plane MKJ.
3.
‹ ›
‹ ›
−
−
Discussion If PQ and RS are different names for the same line, what must be true about
points P, Q, R, and S?
The four points all lie on a common line.
Module 1
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Lesson 1
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Constructing a Copy of a Line Segment
Explain 1
The distance along a line is undefined until a unit distance, such as
1 inch or 1 centimeter, is chosen. You can use a ruler to find the
distance between two points on a line. The distance is the
absolute value of the difference of the numbers on the ruler that
0cm
correspond to the two points. This distance is the length of the
segment determined by the points.
_
In the figure, the length of RS, written RS (or SR), is the distance between R and S.
RS = ⎜4 - 1⎟ = ⎜3⎟ = 3
cm
or
S
R
1
2
3
4
5
SR = ⎜1 - 4⎟ = ⎜-3⎟ = 3 cm
Points that lie in the same plane are coplanar. Lines that lie in the same plane but do not
intersect are parallel. Points that lie on the same line are collinear. The Segment Addition
Postulate is a statement about collinear points. A postulate is a statement that is accepted
as true without proof. Like undefined terms, postulates are building blocks of geometry.
Postulate 1: Segment Addition Postulate
Let A, B, and C be collinear points. If B is
between A and C, then AB + BC = AC.
A
B
C
A construction is a geometric drawing that produces an accurate representation without
using numbers or measures. One type of construction uses only a compass and straightedge.
You can construct a line segment whose length is equal to that of a given segment using
these tools along with the Segment Addition Postulate.
Example 1 Use a compass and straightedge to construct a segment whose
length is AB + CD.
A

B
D
C
Step 1
Use the straightedge to draw a long line segment.
Label an endpoint
X. (See the art drawn in Step 4.)
Step 2 To copy segment AB, open the compass to the distance AB.
A
Step3
Place the compass point on X, and draw an arc. Label the point Y where
the arc and the segment intersect.
Step4
To copy segment CD, open the compass to the distance CD. Place the
compass point on Y, and draw an arc. Label the pointZ
where this second arc and the segment intersect.
X
Y
B
Z
_
XZ is the required segment.
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B
DO NOT
Correcti
B
A
C
D
Step 1 Use the straightedge to draw a long line segment. Label an endpoint X.
Step 2 To copy segment AB, open the compass to the distance AB.
Step 3 Place the compass point on X, and draw an arc. Label the point Y where the
arc and the segment intersect.
Step 4 To copy segment CD, open the compass to the distance CD. Place the compass point on
Y, and draw an arc. Label the point Z where this second arc and the segment intersect.
_
XZ is the required segment.
X
Y
Z
Reflect
4.
Discussion Look at the line and ruler above Example 1. Why does it not matter whether you find the
distance from R to S or the distance from S to R?
The formula to find the distance between the two points involves taking the absolute
value of the difference between the two coordinates R and S, so the distance is always
positive; the order of the coordinates does not matter. From R to S or from S to R, the
coordinates are always 3 units apart.
5.
_
_
In Part B, how can you check
_length of YZ is the same as the length of CD?
_ that the
Use a ruler to measure YZ and CD to see if the lengths are the same.
Your Turn
6.
Use a ruler to draw a segment PQ that is 2 inches _
long. Then use your compass and straightedge to
construct a segment MN with the same length as PQ.
P
Q
M
Explain 2
N
Using the Distance Formula on the
Coordinate Plane
The Pythagorean Theorem states that a 2 + b 2 = c2, where a and b are the lengths of the legs of a
right triangle and c is the length of the hypotenuse. You can use the Distance Formula to apply the
Pythagorean Theorem to find the distance between points on the coordinate plane.
Module 1
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Lesson 1
3/20/14 5:02 PM
The Distance Formula
The distance between two points (x 1, y 1)
and (x 2, y 2) on the coordinate
plane is
y
――――――――
√(x 2 - x 1) 2 + (y 2 - y 1) 2.
(x2, y2)
(x1, y1)
x
Example 2 Determine whether the given segments have
the same length. Justify your answer.
y
4
A
C
E
B
-4
0
x
H
4
D
F
G

-4
_
_
AB and CD
Write the coordinates of the endpoints.
_
Find the length of
AB.
Simplify the expression.
_
Find the length of
CD.
―
A(-4,4), B(1,2), C(2,3), D(4, −2)
――――――――
(2 - 4)
―――― 29
= √5 + (-2) = √―
――――――――
CD = √(4 - 2) + (-2 - 3)
―――― 29
= √2 + (-5) = √―
AB =
√(1 - (-4)) +
2
2
2
2
2
2
2
2
_
_
So, AB = CD = √29 . Therefore,AB and CD have the same length.

_
_
EF and GH
Write the coordinates of the endpoints.
Find the length of
_
EF.
Find the length of
)
―――――――――――
2
2
EF =
=
_
GH.
(
E(-3, 2), F -2 , -3 , G(-2, -4), H
GH =
=
√(
) (
)
(2
, 0
)
-2 - (-3) + -3 - 2
___
√( 1 ) + ( -5 ) = √
2
2
――
26
――――――――――――
2
2
√(
) (0
2 - (-2) +
___
√( 4 ) + ( 4 ) = √
2
2
)
- (-4)
――
32
_
_
EF
and
GH do not have the same length.
EF
≠
GH
So,
. Therefore,
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DO NOT
Correcti
Reflect
7.
Consider how the Distance Formula is related to the Pythagorean
4
Theorem. To use the Distance Formula to____
find the distance from
2
2
U(−3, −1) to V(3, 4), you write UV = √(3 - (-3)) + (4 - (-1)) .
Explain how (3 - (−3)) in the Distance Formula is related to a in the
Pythagorean Theorem and how (4 - (−1)) in the Distance Formula is
y
V
x
-4
U
related to b in the Pythagorean Theorem.
0
4
-4
The Pythagorean Theorem states that a 2 + b 2 = c 2, where a and b are the lengths of the
legs of a right triangle and c is the length of the hypotenuse. Applying this to the right
―――
triangle in the figure, UV = c = √a 2 + b 2 , where a is the length of the horizontal leg of the
triangle, or (3 - (−3)), and b is the length of the vertical leg of the triangle, or (4 - (−1)).
Your Turn
8.
_
_
Determine whether JK and LM have the same length. Justify your
answer.
J(−4, 4), K(−2, 1), L(−1, −4), M(2, −2)
――――――――
2
―
2
JK = (-2 - (-4)) + (1 - 4) = √13
――――――――――
2
2
―
LM = (2 - (-1)) + (-2 - (-4)) = √13
―
―
―
So, JK = LM = √13 . Therefore, JK and LM have the
same length.
√
√
4
R
K
-4
0
P
y
S
x
Q
M
4
L
Explain 3
J
Finding a Midpoint
The midpoint of a line segment is the point that divides the segment into two segments that
have the same length. A line, ray, or other figure that passes through the midpoint of a segment
is a segment bisector .
_
In the figure, _
the tick marks show that PM = MQ. Therefore, M is the midpoint of PQ and
line ℓ bisects PQ.
ℓ
P
M
Q
You can use paper folding as a method to construct a bisector of a given segment
and locate the midpoint of the segment.
Module 1
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Lesson 1
3/20/14 5:02 PM
Example 3 Use paper folding to construct a bisector of each segment.
B
B
B

A
A
A
Step 1
Use a compass
and straightedge
_
to copy AB on a piece of paper.
Fold the paper so that point B
is on top of point A.
Step 2
B
AB
A
Step 3
A
Open the paper. Label the point where the crease intersects the segment as point M.
B
B
A
M
M
A
A
_
_
Point M is the midpoint of AB and the crease is a bisector of AB.

Step 1
Step 2
Step 3
B
_
M straightedge to copy JK on a piece of paper.
Use a compass and
A
Fold the paper so that point K is on top of point
.
Open the paper. Label the point where the crease intersects the segment as
point N.
Point N is the midpoint
Step 4
J
J
―
of JK and the crease is a
bisector
N
K
―
of JK.
Make a sketch of your paper folding construction or attach your folded piece of paper.
Reflect
9.
Explain how you could use paper folding to divide a line segment into four segments
of equal length.
Use paper folding to construct the midpoint of the segment. Then use the same methods
to construct the midpoint of each of the two new segments. The three midpoints divide
the given segment into four segments of equal length.
Module1
GE_MNLESE385795_U1M01L1 11
11
Lesson1
23/03/14 4:56 AM
DO NOT
Correcti
Your Turn
10. Explain how to use a ruler to check your construction in Part B.
Measure each of the segments formed by the bisector. The two segments should each have
a length that is half as long as the given segment.
Finding Midpoints on the Coordinate Plane
Explain 4
You can use the Midpoint Formula to find the midpoint of a segment on the coordinate plane.
The Midpoint Formula
_
The midpoint M of AB with
endpoints A(x 1, y 1)and B(x 2, y 2)
y1 + y2
x1 + x2 _
,
is given by M _
.
2
2
(
y
)
B(x2, y2)
M
A(x1, y1)
( x +2 x , y +2 y )
1
2
1
2
x
Example 4 Show that each statement is true.
_
_
PQ has endpoints P(-4, 1)and Q(2, −3) , then the midpointM of PQ lies in
Quadrant III.
 If
Use the Midpoint Formula to find the midpoint of
(
―
)
-4 + 2 1 + (-3)
M _, _ = M(-1, -1)
2
2
PQ.
Substitute the coordinates, then simplify.
So M lies in Quadrant III, since the x- and y-coordinates are both negative.
_
_
RS has endpoints R(3, 5)and S(−3, −1) , then the midpointM of RS lies on the y-axis.
 If
(
)
(0
5 + -1
3 + -3 _
,
M _
=M
2
2
_
RS.
Substitute the coordinates, then simplify.
, 2
)
So M lies on the y-axis, since the x-coordinate is 0 .
Your Turn
Show that each statement is true.
_
11. If AB has endpoints A(6,_
−3)and B(-6, 3),
then the midpoint M of AB is the origin.
_
12. If JK has endpoints J(7, 0)and K(−5, −4),
_
then the midpoint M of JK lies in Quadrant IV.
6 + (-6) _
-3 + 3
M _
= M(0, 0)
,
2
2
7 + (-5) 0 + (-4)
M _, _ = M(1, -2)
2
2
(
)
(
So M is the origin, since the xand y-coordinates are both 0.
Module1
GE_MNLESE385795_U1M01L1 12
)
Use the Midpoint Formula to find the midpoint of
So M lies in Quadrant IV, since
the x-coordinate is positive and
the y-coordinate is negative.
12
Lesson1
23/03/14 4:57 AM

1.1 Segment Length and Midpoints

Transcription

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