Transcript SEGMENT ADDITION This stuff is AWESOME!

SEGMENT ADDITION
This stuff is
AWESOME!
Can
What
you
about
see anow?
shark?
AB means the line segment with
endpoints A and B.
A
B
AB means the distance between
A and B.
AB = 14 cm
D
G
C
E
E is between C and D.
G is not between C and D.
For one point to be between two other points,
the three points must be collinear.
If Q is between P and R, then PQ + QR = PR.
What does this mean?
Start with a picture:
P
Q
R
If point Q is between points P and R, then the distance between P and
plus the distance between Q and R is equal to the distance between P
If PQ + QR = PR, then Q is between P and R.
What does this mean?
If the measure of segment PQ plus the measure of segment QR is equal to
the measure of segment PR, then point Q must be between points P and R.
12
P
3
PR = 15
R
Q
14
P
Q
15
3
R
COLORED NOTE CARD
Segment Addition Postulate
#2
If Q is between P and R, then PQ + QR = PR.
P
Q
R
If PQ + QR = PR, then Q is between P and R.
N is between L and P. LN = 14 and PN = 12. Find LP.
L
14
N
12
P
Q is between R and T. RT = 18 and QR = 10. Find QT.
10
R
Q
18
T
Find MN if N is between M and P, MN = 3x + 2,
NP = 18, and MP = 5x.
M
3x + 2
N
18
P
5x
3x + 2 + 18 = 5x
3x + 20 = 5x
-3x
-3x
20 = 2x
2
2
10 = x
MN = 3 (10 ) + 2
MN = 32
THE DISTANCE AND MIDPOINT FORMULAS
Investigating Distance:
1 Plot A(2,1) and B(6,4) on a coordinate
plane. Then draw a right triangle that
y
has AB
its hypotenuse.
2 Find
andaslabel
the coordinates of the
vertex C.
3 Find the lengths of the legs
ABC.
of
B (6, 4)
AB
5
y4B –3 y1A
A (2, 1)
4 Use the Pythagorean theorem to find AB.
Remember:
C (6, 1)
x
a2 + b2 = c2
42 + 32 =
c2
16 + 9 =
c2
x6B –4 x2A
16 + 9 =
c
25 =
c
AB = 5
Finding the Distance Between Two Points
The steps used in the investigation can be used to develop
a general formula for the distance between two points A(x
1, y 1) and B(x 2, y 2).
B (x , y )
Using the Pythagorean theorem
y
a2 + b = c
2
d
You can2write
the equation
y –y
2
2
2
(x 2 – x 1) 2 + (y 2 – y 1) 2 = d 2
1
A (x 1, y 1 )
Solving this for d
produces the
distance
formula.
THE DISTANCE
FORMULA
The distance d between the points (x 1, y 1) and
(x 2, yd2) =is (x 2 – x 1) 2 + (y 2
– y 1) 2
x2 – x1
C (x 2, y 1 )
x
Finding the Distance Between Two Points
Find the distance between (1, 4) and (–2, 3).
SOLUTI
ONfind the distance, use the distance formula.
To
Write the distance formula.
d = (x 2 – x 1) 2 + (y 2
– y 1) 2
= –2
(x–2 –1 x 1) 23+– (y
4 2 – y 1) 2Substitute.
=
10
 3.16
Simplify.
Use a calculator.
Applying the Distance Formula
A player kicks a soccer ball that is 10 yards from a sideline and 5 yards from a goal line.
The ball lands 45 yards from the same goal line and 40 yards from the same sideline. How
far was the ball kicked?
SOLUTI
The ball is kicked from the point (10, 5), and
ON
lands at the point (40, 45). Use the distance
formula.
d = (40 – 10) 2 +
(45 – 5) 2
= 900 + =
1600
2500
= 50
The ball was kicked 50 yards.
Finding the Midpoint Between Two Points
The midpoint of a line segment is the point on the segment that is equidistant from its endpoints. The midpoint between two points is the midpoint of the line segment connecting
them.
THE MIDPOINT FORMULA
(
x + x2 y1 +
The midpoint between the points (x 1, y 1) and (x 1
,
2
y2
,
y
)
is
2
2
2
)
Finding the Midpoint Between Two Points
Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the result.
x1 + x2 y1 +
SOLUTI Remember, the midpoint formula
( is ,
2
y2
ON
.
5
–2 + 4 3 + 2
2 5
2
1
( 2 , 2 ) =( 2 , 2 ) =( , 2 )
5
The midpoint (is1 2 ,)
.
)
Finding the Midpoint Between Two Points
Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the result.
CHECK
(–2, 3)
From the5graph, you can see
1 2)
(
that the point
,
(1, 5 )
2
(4, 2)
appears halfway between (–
2, 3) and (4, 2). You can also
use the distance formula to
check that the distances from
the midpoint to each given
Applying the Midpoint Formula
You are using computer software to design a video game. You want to place a buried
treasure chest halfway between the center of the base of a palm tree and the corner of a
large boulder. Find where you should place the treasure chest.
SOLUTI
(25, 175)
ON
1 Assign coordinates to the
locations of the two landmarks.
(112.5, 125)
The center of the palm tree is at
(200, 75). The corner of the
at (25, 175).
Use the is
midpoint
formula to find
2 boulder
25
the point 175
that+is halfway between
25 +two
200landmarks.
75
225 0
the
( 2 , 2 ) = ( 2 , 2 ) = (112.5, 125)
(200, 75)
ANGLES
You will learn to classify
angles as acute,
obtuse, right, or
straight.
What is an angle?

Two rays that share the same endpoint
form an angle. The point where the rays
intersect is called the vertex of the
angle. The two rays are called the sides
of the angle.
 Here
are some examples
of angles.

We can identify an angle by using a point on
each ray and the vertex. The angle below
may be identified as angle ABC or as angle
CBA; you may also see this written as
<ABC or as <CBA. The vertex point is
always in the middle.
Angle Measurements


We measure the size of an angle using
degrees.
Here are some examples of angles and
their degree measurements.
Acute Angles


An acute angle is an angle measuring
between 0 and 90 degrees.
The following angles are all acute angles.
Obtuse Angles


An obtuse angle is an angle measuring
between 90 and 180 degrees.
The following angles are all obtuse.
Right Angles


A right angle is an angle measuring 90
degrees.
The following angles are both right angles.
Straight Angle

A straight angle is 180 degrees.
Adjacent,
Vertical, Linear Pair
Supplementary, and
Complementary Angles
Adjacent angles are “side by side”
and share a common ray.
45º
15º
These are examples of adjacent
angles.
80º
45º
35º
55º
85º
20º
130
º
50º
These angles are NOT adjacent.
100
º
50º
35º
35º
55º
45º
When 2 lines intersect, they make
vertical angles.
75º
105
º
105
º
75º
Vertical angles are opposite one
another.
75º
105
º
105
º
75º
Vertical angles are opposite one
another.
75º
105
º
105
º
75º
Vertical angles are congruent (equal).
150
30º
º
150
30º
º
Linear Pair are adjacent angles that add
to be 180 degrees.
75º
105
º
105
º
75º
Supplementary angles add up to
180º.
40º
120
º
60º
Linear Pair: Adjacent
and Supplementary
Angles
140
º
Supplementary Angles
but not Adjacent
Complementary angles add up to
90º.
30º
40º
60º
Adjacent and
Complementary Angles
50º
Complementary Angles
but not Adjacent
1.6 Classify Polygons
Identifying Polygons




Formed by three or more line segments
called sides.
It is not open.
The sides do not cross.
No curves.
NOT
POLYGONS
POLYGONS
Terms

Convex: a polygon is convex if no line that
contains a side of the polygon contains a point in
the interior of the polygon.
•Concave: a polygon that is nonconvex.
Classifying Polygons
Number of Type of Polygon
Sides
Number of
Side
Type of Polygon
3
Triangle
8
Octagon
4
Quadrilateral
9
Nonagon
5
Pentagon
10
Decagon
6
Hexagon
12
Dodecagon
7
Heptagon
n
n-gon
Definitions




n-gon: a polygon with n number of sides.
Equilateral: a polygon whose sides are all
congruent.
Equiangular: a polygon whose angles are
all congruent.
Regular: a polygon whose sides are
equilateral and whose angles are
equiangular.
Determine if the figure is a polygon. If yes, state
whether it is convex or concave
Yes, conclave
Yes, convex
This figure is equilateral because all sides are the congruent
It is also equiangular because all angles are congruent.
Therefore this is a regular pentagon.
1.7 Find Perimeter,
Circumference, and Area
Formulas
Square
Rectangle
Area: s2
Area: lw
w
Perimeter: 4s
l
s
Triangle
Circle
Area: bh
a h
b
c
2
Perimeter: a + b + c
r
d
Perimeter: 2l + 2w
Area: πr2
Circumference: 2πr
Find perimeter and area of the
figure below.
9 ft
12 ft
A=lxw
P = 2l + 2w
A = 108 ft2
P = 42 ft
Find the approximate area
and circumference of the
figure below
18 in
A = πr2
C = 2πr or dπ
A = 254.3 in2
C = 56.5 in
A triangle ABC has vertices
A(2,5), B(4,1), and C(8,3).
What is the approximate
perimeter of ΔABC?





Hint use the distance formula for AB,
AC, and BC (d = √[(x2-x1)2 +(y2-y1)2])
AB = 4.47
AC = 6.32
BC = 4.47
P = 15.26