Pg. 1

Geometry Summer Packet

RADICALS

Radical expressions contain numbers and/or variables under a radical sign, . A radical

sign tells you to take the square root of the value under the symbol. The radicand is the expression under the radical sign. EXAMPLE 1. SIMPLIFY

a) 18 b) 140

2 3 3

2 3 3

3 2

2 2 5 7

2 2 5 7

2 5 7

2 35

PRACTICE

1) 75 2) 80 3) 280 4) 500

Product Property of Square Roots: For any numbers a and b where

0 and 0, .a b ab a b

EXAMPLE 2. SIMPLIFY

a) 5 35

5 35

5 5 7

5 7

PRACTICE

1) 3 7 49 2) 7 30 6

Quotient Property of Square Roots: For any numbers a and b where

0 and 0, .a a

a bb b

Pg. 2

EXAMPLE 3. SIMPLIFY

a) 34

25 b)

56

7

34

25

34

5

56

7

8

2 2 2

2 2

PRACTICE

1) 20

5 2)

4

16

Rationalizing the Denominator: This method may be used to remove or eliminate radicals from the denominator of a fraction. EXAMPLE 4. SIMPLIFY

a) 5

3 b)

7

12

5 3

3 3

15

9

15

3

7

2 2 3

7 3

2 3 3

21

2 9

21

2 3

21

6

PRACTICE

1) 3

7 2)

2

3 3)

11

32

Pg. 3

THE PYTHAGOREAN THEOREM If a and b are the measure of the legs of a right triangle and c is the measure of the

hypotenuse, then 2 2 2.a b c

c

b

a

You use the Pythagorean Theorem to find the length of the missing side of a right triangle. EXAMPLE 1. FIND THE LENGTH OF THE MISSING SIDE OF A RIGHT TRIANGLE IF a = 12 AND b = 5.

2 2 2

2

2

2

12 5

144 25

169

169

13

c

c

c

c

c

Yet, you only keep +13 because c is a distance measure. PRACTICE 1) a = 4 and b = 3 2) b = 8 and c = 10 3) a = 7 and b = 24

Pg. 4

DISTANCE FORMULA This formula is used to find the distance d between any two points with coordinates (x1 , y1)

and (x2 , y2) is given by the following formula: 2 2

2 1 2 1 .d x x y y

EXAMPLE 1. FIND THE DISTANCE BETWEEN THE POINTS WITH COORDINATES (3, 5) AND (6, 4).

let 1 1 2 2, 3, 5 and , 6, 4 x y x y

2 2

6 3 4 5 d

2 2

3 1

9 1

10

PRACTICE 1) Find the distance between the points with coordinates (-3, 4) and (-1, -5). 2) Find the distance between the points with coordinates (-3, 4) and (5, 2).

Pg. 5

THE COORDINATE PLANE When we graph ordered pairs (points), we have an x and a y value to graph. The ordered

pair is represented by , .x y We graph this coordinate on a Coordinate Plane. The x

represents the right or left movement, which is counted first. Then, the y represents the up or down movement, which is counted second. The point will end up in one of the four Quadrants which are sections of the Coordinate Plane. Usually, these Quadrants are labeled with Roman Numerals.

EXAMPLE 1. PLOT THE POINTS AND IDENTIFY THE QUADRANT OR AXIS.

a) (2,5) is in Quadrant I b) ( 2,3) is in Quadrant II

c) ( 4, 1) is in Quadrant III d) 5, 1 is in Quadrant IV

e) (0, 2) is on the y-axis f) (2,0) is on the x-axis

Pg. 6

PRACTICE

1) ( 3, 2) is in Quadrant _____ 2) (0, 3) is on the _____ axis

3) (4,1) is in Quadrant _____ 4) ( 3, 2) is in Quadrant _____

5) (5,0) is on the _____ axis 6) (5, 2) is in Quadrant _____

Pg. 7

SLOPE The ratio, m, of the rise to the run as you move from one point to another along a line.

rise

runm

rise: vertical change run: horizontal change

run

rise

Determining Slope Given Two Points: Given the coordinates of two points 1 1,x y and

2 2,x y on a line the slope can be found as follows: 2 1

2 1

.y y

mx x

EXAMPLE 1. DETERMINE THE SLOPE OF THE LINE THAT PASSES THROUGH (2, -5) AND (7, -10).

let 1 1 2 2, 2, 5 and , 7, 10 x y x y

10 5

7 2m

5

5

1

PRACTICE 1) Find the slope of the line that passes through (4, 6) and (10, -3). 2) Find the slope of the line that passes through (2, -1) and (5, -3).

Pg. 8

LINEAR EQUATIONS

Point-Slope Form: For a given point 1 1,x y on a nonvertical line having slope m, the point-slope

form is as follows 1 1 .y y m x x

EXAMPLE 1. WRTIE THE POINT-SLOPE FORM OF AN EQUATION FOR A LINE THAT

PASSES THROUGH (-3, 5) AND HAS A SLOPE OF 3

.4

let 1 1

3, 3, 5 and

4

x y m

3

5 34

y x

PRACTICE

1) Write the point-slope form of an equation for a line that passes through (3, 8) and has a slope of 2.

2) Write the point-slope form of an equation for a line that passes through (-6, 1) and has a

slope of 2

.5

Standard Form: The standard form of a linear equation is Ax+By=C, where A, B, and C are

integers, 0,A and A and B are not both zero.

EXAMPLE 2. WRITE 5

5 24

y x

IN STANDARD FORM.

5

4( 5) 4 24

y x

4 20 5( 2)

4 20 5 10

4 5 10 20

4 5 10

5 4 10

y x

y x

y x

y x

x y

PRACTICE

1) Write 3

3 14

y x

in standard form.

2) Write 3 2 9y x in standard form.

Pg. 9

Slope-Intercept Form: Given the slope m and the y-intercept b of a line, the slope-intercept form

of an equation of the line is .y mx b

EXAMPLE 3. WRITE AN EQUATION OF A LINE IN SLOPE-INTERCEPT FORM IF THE

LINE HAS A SLOPE OF 2

3 AND A Y-INTERCEPT OF 6.

let m2

and 63

b

2

63

y x

PRACTICE 1) Write an equation of a line in slope-intercept form if the line has a slope of 3 and a y-intercept of 5.

2) Write an equation of a line in slope-intercept form if the line has a slope of 1

4 and a

y-intercept of -10.

EXAMPLE 4. FIND THE SLOPE AND Y-INTERCEPT OF THE GRAPH OF 5 3 6.x y

Turn the equation into slope-intercept form.

3 5 6

3 5 6

3 3 3

52

3

y x

y x

y x

m5

3 and 2b

PRACTICE

1) Find the slope and y-intercept of the graph of 5 4 10.x y

2) Find the slope and y-intercept of the graph of 1

4 2.3

x y

Pg. 10

ABSOLUTE VALUE An absolute value of a number is the distance the number is from zero. Therefore, the solution of an absolute value is always positive due to the fact that it represents a distance.

The symbol for absolute value is .

EXAMPLE 1. FIND THE ABSOLUTE VALUE.

a) 5 b) 5 c) 5 3 d) 10 18

=5 =5 = 2 = 8

=2 =8 PRACTICE

1) 7 2) 7 3) 12 8 4) 11 23

Pg. 11

FRACTIONS There are three simple steps to add or subtract fractions:

Step 1: Make sure the denominators are the same. If they are not the same, find the Least Common Denominator.

Step 2: Add or subtract the numerators and put the answer over the denominator. Step 3: Simplify the fraction (if needed).

EXAMPLE 1. SIMPLIFY.

a) 1 1

4 4 b)

8 1

6 2 c)

2 3

5 2

=1 1

4

=

8 1 3

6 2 3

=

2 2 3 5

5 2 2 5

=2

4 =

8 3

6 6 =

4 15

10 10

=1

2 =

8 3

6

=

4 15

10

=5

6 =

19

10

PRACTICE.

1) 7 2

6 6 2)

7 2

9 3 3)

5 4

7 5

4)

7 3

9 4

There are three simple steps to multiply fractions

Step 1: Multiply the numerators.

Step 2: Multiply the denominators.

Step 3: Simplify the fraction (if needed). EXAMPLE 2. SIMPLIFY.

a) 1 1

*4 4

b)

8 1*

6 2 c)

2 3*

5 2

=1*1

4 * 4

=

8 *1

6 * 2 =

2 * ( 3)

5 * 2

=1

16

=

8

12 =

3

5

=2

3

PRACTICE.

5) 3 12

*9 7

6)

8 5*

3 2 7)

4 6*

11 5

Pg. 12

There are two simple steps to divide fractions

Step 1: Multiply the first fraction by the reciprocal of the second fraction.

Step 2: Simplify the fraction (if needed). EXAMPLE 3. SIMPLIFY.

a) 2 1

3 4

b)

1 3

2 6 c)

15

9

=2 4

*3 1

=

1 6*

2 3 =

1 1*

9 5

=2 * 4

3*1

=

1* 6

2 * 3 =

1*1

9 * 5

=8

3

=

6

6 =

1

45

=1 PRACTICE.

8) 6 4

7 5

9)

2( 9)

6 10)

8 4

11 7

Pg. 13

SOLVING EQUATIONS EXAMPLE 1. SOLVE EACH EQUATION FOR THE VARIABLE. a) 6 2m b) 2 8x

2 6

4

m

m

8 2

6

x

x

6

1 1

6

x

x

PRACTICE 1) 2 7g 2) 9 5s

3) ( 6) 5a 4) ( 44) 61d

EXAMPLE 2. SOLVE EACH EQUATION FOR THE VARIABLE.

a) 5 35p b) 759

c

5 35

5 5

7

p

p

( 59) 7 59

59

413

c

c

PRACTICE 1) 3 24x 2) 6 2a

3) 6314

f 4) 84

97

x

EXAMPLE 3. SOLVE EACH EQUATION FOR THE VARIABLE. a) 6 7 8 13x x b) 8(4 9 ) 7( 2 11 )x x

7 13 8 6

20 2

20 2

2 2

10

x x

x

x

x

32 72 14 77

77 72 32 14

5 18

5 18

5 5

18

5

x x

x x

x

x

x

PRACTICE 1) 17 3 21 2x x 2) 7 3 2 4(2 )x x x

3) 1

7(2 5) ( 12 30)2

x x 4) 6 3( 3 )3

xx

Pg. 14

SOLVING SYSTEMS OF EQUATIONS Sometimes when solving equations there are two or more variables we need to solve for. Most of the time in Geometry, you will be given two equations with two variables which is known as a system of equations. You are looking for the coordinate, ( , ),x y that works for

both equations. In Algebra, you should have learned three different methods for solving systems. The easiest methods are to use Substitution or Addition/Elimination. EXAMPLE 1. SOLVE THE SYSTEM USING SUBSTITUTION.

a) 2 3

4 4 8

x y

x y

First, you need to solve for one of the variables, that means get one variable by itself. Sometimes this step is already done for you. The obvious choice would be to either solve for x or y in the first equation. Let’s solve for y in the first equation.

3 2y x

Now, go the second equation and substitute 3 2x in place of the variable y. 4 4(3 2 ) 8x x

Then, solve for x.

4 12 8 8

4 12 8

4 8 12

4 4

4 4

4 4

1

x x

x

x

x

x

x

Finally, substitute 1 in place of the variable x in either of the two original equations. It is usually easiest to substitute the value into the equation that you simplified.

3 2(1)

3 2

1

y

y

y

Thus, your answer is 1,1 .

Pg. 15

EXAMPLE 2. SOLVE THE SYSTEM USING ADDITION/ELIMINATION.

a) 2 3

4 4 8

x y

x y

First, you need to add both equations together, so that either the x or the y cancels out. If a variable won’t cancel when adding the original equations, you need to change the equations. You do this by multiply one or both of the equations by numbers. In this case, you can either turn the 2x in the top equation into -4x, or you can turn the y in the top equation into -4y. Let’s cancel the x values. Thus, you need to multiply the entire top equation by -2.

2 3 ( 2)x y

4 2 6x y

Now, add the two equations together. The x will cancel out in both equation.

4 2 6

4 4 8

x y

x y

0 2 2

2 2

x y

y

Then, solve for the remaining variable, y.

2 2

2 2

2

y

y

Once you have solved for one variable, substitute the value of the variable into either of the two original equations. Let’s go back to the first equation and substitute 1 for y. 2 1 3x Finally, solve for x.

2 3 1

2 2

2 2

2 2

1

x

x

x

x

Thus, your answer is 1,1 .

PRACTICE. SOLVE EACH SYSTEM OF EQUATIONS USING EITHER METHOD.

1) 4

3 2

y x

x y

2) 2 8

2 9

x y

x y

Pg. 16

Geometry Summer Packet – Answers

RADICALS EXAMPLE 1 PRACTICE

1) 5 3 2) 4 5 3) 2 70 4) 10 5

EXAMPLE 2 PRACTICE

1) 21 7 2) 42 5

EXAMPLE 3 PRACTICE

1) 2 2) 1

2

EXAMPLE 4 PRACTICE

1) 21

7 2)

6

3 3)

22

8

THE PYTHAGOREAN THEOREM

EXAMPLE 1 PRACTICE 1) 5c 2) 6a 3) 25c

DISTANCE FORMULA EXAMPLE 1 PRACTICE

1) 85 2) 2 17

THE COORDINATE PLANE

EXAMPLE 1 PRACTICE 1) III 2) y 3) I 4) II 5) x 6) IV

Pg. 17

SLOPE EXAMPLE 1 PRACTICE

1) 3

2m

2)

2

3m

LINEAR EQUATIONS

EXAMPLE 1 PRACTICE

1) 8 2( 3)y x 2) 2

1 ( 6)5

y x

EXAMPLE 2 PRACTICE 1) 3 4 9x y 2) 2 15x y

EXAMPLE 3 PRACTICE

1) 3 5y x 2) 1

104

y x

EXAMPLE 4 PRACTICE

1) 5 5

and 4 2

m b

2) 5 5

and 4 2

m b

ABSOLUTE VALUE

EXAMPLE 1 PRACTICE 1) 7 2) 7 3) 4 4) 12

FRACTIONS EXAMPLE 1 PRACTICE

1) 5

6 2)

13

9 3)

3

35 4)

1

36

EXAMPLE 2 PRACTICE

5) 4

7

6)

20

3 7)

24

55

EXAMPLE 3 PRACTICE

8) 15

14 9)

1

27

10)

14

11

SOLVING EQUATIONS

EXAMPLE 1 PRACTICE 1) 9y 2) 14s 3) 11a 4) 17d

EXAMPLE 2 PRACTICE

1) 8x 2) 1

3a 3) 882f 4) 8148x

Pg. 18

EXAMPLE 3 PRACTICE

1) 4x 2) 15x 3) 5

2x 4)

9

13x

SOLVING SYSTEMS OF EQUATIONS

EXAMPLE 1 PRACTICE

1) 3, 7 2) 5, 2