SEGMENT ADDITION

This stuff is AWESOME!

Can you see a shark?What about now?

AB means the line segment with endpoints A and B.

AB means the distance between A and B.

A B

AB = 14 cm

C

D

EG

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 RQ

If point Q is between points P and R, then the distance between P and Q plus the distance between Q and R is equal to the distance between P and R.

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.

P RQ

P RQ

PR = 1512 3

14 3

15

COLORED NOTE CARDSegment Addition Postulate #2

If Q is between P and R, then PQ + QR = PR.

If PQ + QR = PR, then Q is between P and R.

P RQ

N is between L and P. LN = 14 and PN = 12. Find LP.

L PN

Q is between R and T. RT = 18 and QR = 10. Find QT.

R TQ

14 12

18

10

Find MN if N is between M and P, MN = 3x + 2,NP = 18, and MP = 5x.

M PN3x + 2 18

5x

3x + 2 + 18 = 5x 3x + 20 = 5x -3x -3x 20 = 2x 2 2 10 = x

MN = 3 (10 ) + 2MN = 32

yB – yA

xB – xA

THE DISTANCE AND MIDPOINT FORMULAS

Investigating Distance:

2 Find and label the coordinates of the vertex C.

3

x

y

C (6, 1)

1 Plot A(2,1) and B(6,4) on a coordinate plane. Then draw a right triangle that has AB as its hypotenuse.

Remember:

a 2 + b

2 = c 2

4 – 1

6 – 2

AB

4

3

4 2 + 3

2 = c

2

16 + 9 = c

2

16 + 9 = c25 = c

5

4Use the Pythagorean theorem to find AB.

AB = 5

B (6, 4)

A (2, 1)

Find the lengths of the legs ofABC.

Finding the Distance Between Two Points

Using the Pythagorean theorem

(x 2 – x 1) 2 + ( y 2 – y 1)

2 = d 2

THE DISTANCE FORMULA

The distance d between the points (x 1, y 1) and

(x 2, y 2) isd = (x 2 – x 1)

2 + ( y 2 – y 1)

2

Solving this for d

produces the

distance formula.

You can write the equation

a 2 + b

2

= c

2

x 2 – x 1

y2 – y1

d

x

y

C (x 2, y 1 )

B (x 2, y 2 )

A (x 1, y 1 )

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).

Finding the Distance Between Two Points

Find the distance between (1, 4) and (–2, 3).

d = (x 2 – x 1) 2 + ( y 2

– y 1) 2

= 10

3.16

To find the distance, use the distance formula.

Write the distance formula.

Substitute.

Simplify.

Use a calculator.

SOLUTI

ON

= (x 2 – x 1) 2 + ( y 2 – y 1)

2–2 – 1 3 – 4

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?

The ball is kicked from the point (10, 5), and

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.

SOLUTI

ON

Finding the Midpoint Between Two Points

The midpoint of a line segment is the point on the segment that is equidistant from its end-

points. The midpoint between two points is the midpoint of the line segment connecting

them.

THE MIDPOINT FORMULA

The midpoint between the points (x 1, y 1) and (x

2, y 2) is

x 1 + x 2

2( )y 1 + y

2

2

,

Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the result.

SOLUTI

ON –2 + 4

2( )3 + 2

2,22( )

52,= 1( )

52,=

The midpoint is , .

1( )52

x 1 + x 2

2( )y 1 + y

2

2

,Remember, the midpoint formula is

.

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

From the graph, you can see

that the point ,

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

point are equal.

( )152

Finding the Midpoint Between Two Points

(1, )52

(–2, 3)

(4, 2)

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

ON Assign coordinates to the locations of the two landmarks. The center of the palm tree is at (200, 75). The corner of the boulder is at (25, 175). Use the midpoint formula to find the point that is halfway between the two landmarks.

1

2

25 + 2002( )

175 + 752,

2252( )

2502,= = (112.5, 125)

(25, 175)

(200, 75)

(112.5, 125)

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 PairSupplementary, and

Complementary Angles

Adjacent angles are “side by side” and share a common ray.

45º15º

These are examples of adjacent angles.

55º

35º

50º130º

80º 45º

85º20º

These angles are NOT adjacent.

45º55º

50º100º 35º

35º

When 2 lines intersect, they make vertical angles.

75º

75º

105º

105º

Vertical angles are opposite one another.

75º

75º

105º

105º

Vertical angles are opposite one another.

75º

75º

105º

105º

Vertical angles are congruent (equal).

30º150º

150º

30º

Linear Pair are adjacent angles that add to be 180 degrees.

75º

75º

105º

105º

Supplementary angles add up to 180º.

60º120º

40º

140º

Linear Pair: Adjacent and Supplementary

Angles

Supplementary Anglesbut not Adjacent

Complementary angles add up to 90º.

60º

30º 40º50º

Adjacent and Complementary Angles

Complementary Anglesbut 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.

POLYGONS

NOT 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 PolygonsNumber of

SidesType of Polygon Number of

SideType 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 congruentIt is also equiangular because all angles are congruent.Therefore this is a regular pentagon.

1.7 Find Perimeter, Circumference, and Area

Formulas

Square Rectangle

Triangle Circles

Area: s2

Perimeter: 4s

l

wArea: lw

Perimeter: 2l + 2w

h

b

Area: bh

2

Perimeter: a + b + c

a cdr

Area: πr2

Circumference: 2πr

Find perimeter and area of the figure below.

9 ft

12 ft

A = l x w

P = 2l + 2w

A = 108 ft2

P = 42 ft

Find the approximate area and circumference of the figure below

A = πr2

C = 2πr or dπ

A = 254.3 in2

C = 56.5 in

18 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