Mathematics exam 3 doc

Similar Polygons If two polygons are similar, their corresponding angles are equal and the ratios of the corresponding sides are in proportion.. Example These two polygons are similar be

the sum of its interior angles will equal 180 + 180 = 360

degrees

m∠a + m∠b + m∠c + m∠d = 360°

Interior Angles

To find the sum of the interior angles of any polygon, use

this formula:

S = 180(x − 2)°, with x being the number of

polygon sides

Example

Find the sum of the angles in this polygon:

S = (5 − 2) × 180°

S = 3 × 180°

S = 540°

Exterior Angles

Similar to the exterior angles of a triangle, the sum of the

exterior angles of any polygon equals 360 degrees.

Similar Polygons

If two polygons are similar, their corresponding angles

are equal and the ratios of the corresponding sides are in

proportion

Example

These two polygons are similar because their angles are equal and the ratios of the correspon-ding sides are in proportion

Parallelograms

A parallelogram is a quadrilateral with two pairs of

par-allel sides

In the figure above, line AB || CD and BC || AD.

A parallelogram has:

■ opposite sides that are equal (AB = CD and BC = AD)

■ opposite angles that are equal (m∠a = m∠c and m∠b = m∠d)

■ consecutive angles that are supplementary (m∠a + m∠b = 180°, m∠b + m∠c = 180°, m∠c + m∠d

= 180°, m∠d + m∠a = 180°)

S PECIAL T YPES OF P ARALLELOGRAMS

■ A rectangle is a parallelogram that has four right

angles

D A

AB = CD

D A

60 ° 10

4

6

18

120°

60°

120 ° 5

2

3

9

b

c

d e

a

b

d

■ A rhombus is a parallelogram that has four equal

sides

■ A square is a parallelogram in which all angles are

equal to 90 degrees and all sides are equal to each

other

D IAGONALS

In all parallelograms, diagonals cut each other into two

equal halves

■ In a rectangle, diagonals are the same length

■ In a rhombus, diagonals intersect to form

90-degree angles

■ In a square, diagonals have the same length and intersect at 90-degree angles

 S o l i d F i g u r e s , P e r i m e t e r,

a n d A r e a

The GED provides you with several geometrical formu-las These formulas will be listed and explained in this section It is important that you be able to recognize the figures by their names and to understand when to use which formulas Don’t worry You do not have to mem-orize these formulas They will be provided for you on the exam

To begin, it is necessary to explain five kinds of measurement:

1 Perimeter

The perimeter of an object is simply the sum

of all of its sides

2 Area

Area is the space inside of the lines defining the shape

3 Volume

Volume is a measurement of a three-dimensional object such as a cube or a

= Area

6

7

4

10

Perimeter = 6 + 7 + 4 + 10 = 27

B

C

A

D

AC = DB

and

AC DB

BD AC

AC = DB

D

C B

A

AB = BC = CD = DA

m∠A = m∠B = m∠C = m∠D

B A

AB = BC = CD = DA

rectangular solid An easy way to envision

vol-ume is to think about filling an object with

water The volume measures how much water

can fit inside

4 Surface Area

The surface area of an object measures the

area of each of its faces The total surface area of

a rectangular solid is double the sum of the areas

of the three faces For a cube, simply multiply the

surface area of one of its sides by six

5 Circumference

Circumference is the measure of the distance

around the outside of a circle

 C o o r d i n a t e G e o m e t r y

Coordinate geometry is a form of geometrical operations in relation to a coordinate plane A coordinate plane is a grid

of square boxes divided into four quadrants by a

hori-zontal (x) axis and a vertical (y) axis These two axes inter-sect at one coordinate point, (0,0), the origin A coordinate point, also called an ordered pair, is a specific point on the

coordinate plane with the first number, or coordinate, representing the horizontal placement and the second number, or coordinate, representing the vertical

place-ment Coordinate points are given in the form of (x,y).

Graphing Ordered Pairs

The x-coordinate

■ The x-coordinate is listed first in the ordered pair

and it tells you how many units to move to either

the left or to the right If the x-coordinate is positive, move to the right If the x-coordinate is

negative, move to the left

The y-coordinate

■ The y-coordinate is listed second and tells you

how many units to move up or down If the

y-coordinate is positive, move up If the y-coordinate is negative, move down.

Example

Graph the following points: (2,3), (3,−2), (−2,3), and (−3,−2)

Notice that the graph is broken up into four quadrants with one point plotted in each one

(−2,3) (2,3)

(−3,−2) (3,−2)

Circumference

4

4

Surface area of front side = 16 Therefore, the surface area

of the cube = 16 x 6 = 96.

This chart indicates which quadrants contain which

ordered pairs based on their signs:

Lengths of Horizontal and Vertical

Segments

Two points with the same y-coordinate lie on the

same horizontal line and two points with the same

x-coordinate lie on the same vertical line The length of

a horizontal or vertical segment can be found by taking

the absolute value of the difference of the two points or

by counting the spaces on the graph between them

Example

Find the lengths of AB  and line BC

Solution:

| 2 − 7 | = 5 = AB

| 1 − 5 | = 4 = BC

Midpoint

To find the midpoint of a segment, use the following

for-mula:

Midpoint x = x1 +

2

x2

 Midpoint y = y1 +

2

y2



Example

Find the midpoint of line segment AB .

Solution:

Midpoint x = 1 +25 = 62= 3

Midpoint y = 2 + 120= 122= 6

Therefore the midpoint of AB is (3,6)

(5,10)

Midpoint

(1,2)

B

A

(2,1)

(7,5) C

B A

Sign of

Points Coordinates Quadrant

(2,3) (+,+) I

(–2,3) (–,+) II

(–3,–2) (–,–) III

(3,–2) (+,–) IV

The slope of a line measures its steepness It is found

by writing the change in the y-coordinates of any two

points on the line, over the change of the corresponding

x-coordinates (This is also known as the rise over the

run.) The last step is to simplify the fraction that results.

Example

Find the slope of a line containing the points

(3,2) and (8,9)

Solution:

98−−23= 75

Therefore, the slope of the line is 75

Note: If you know the slope and at least one point on a

line, you can find the coordinates of other points on the

line Simply move the required units determined by the

slope In the last example, from (8,9), given the slope 75,

move up seven units and to the right five units Another

point on the line, thus, is (13,16)

I MPORTANT I NFORMATION ABOUT S LOPE

■ A line that rises from left to right has a positive slope and a line that falls from left to right has a negative slope

■ A horizontal line has a slope of 0, and a vertical line does not have a slope at all—it is undefined

■ Parallel lines have equal slopes

■ Perpendicular lines have slopes that are negative reciprocals

(3,2)

(8,9)

BA S I C P R O B L E M S O LV I N G in mathematics is rooted in whole number math facts, mainly addition

facts and multiplication tables If you are unsure of any of these facts, now is the time to review Make sure to memorize any parts of this review that you find troublesome Your ability to work with numbers depends on how quickly and accurately you can do simple mathematical computations

 O p e r a t i o n s

Addition and Subtraction

Addition is used when you need to combine amounts The answer in an addition problem is called the sum or the total It is helpful to stack the numbers in a column when adding Be sure to line up the place-value columns

and to work from right to left

Number Operations and Number Sense

A GOOD grasp of the building blocks of math will be essential for

your success on the GED Mathematics Test This chapter covers the basics of mathematical operations and their sequence, variables, inte-gers, fractions, decimals, and square and cube roots

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