Geometry 263 Prerequisites page 1

Name_________________________________

Dear Incoming Geometry Student,

Listed below are fifteen skills that we will use throughout Geometry 263. You have likely learned and

practiced these skills in previous math classes. Although we will briefly review a skill in class the first time

we use it, you are ultimately responsible for the mastery of the skill.

The following pages describe the target skills and provide some worked-out examples along with an

opportunity for you to practice the skill on your own (answers are on the last three pages of the packet).

You should work out the problems by yourself. This packet is not mandatory, but we highly recommend

that you work enough problems from each section to reassure yourself that you have mastered the skill.

If you need help please visit with me outside of class, or visit the Titan Learning Center (in the GBS

library) to meet with a tutor. There are also many online options such as Khan Academy.

Numbers

Target N1: I can multiply a fraction and a whole number without a calculator.

Target N2: I can simplify the square root of a whole number without a calculator.

Target N3: I can multiply square roots without a calculator.

Target N4: I can add square roots without a calculator.

Algebra

Target A1: I can multiply polynomials (monomials and binomials) without a calculator.

Target A2: I can solve linear equations without a calculator.

Target A3: I can solve proportions, including those that lead to x n2= (where n is a whole number),

without a calculator.

Target A4: I can solve a system of linear equations by the elimination method.

Target A5: I can solve a quadratic equation with factoring and without a calculator.

Target A6: I can solve a quadratic equation with the quadratic formula.

Geometry

Target G1: I can determine the coordinates of a point in the xy-coordinate plane.

Target G2: Given two ordered pairs in the xy-plane, I can calculate the distance between the points,

the slope of the line connecting the points, and the coordinates of the midpoint.

Target G3: Given two side lengths of a right triangle (a triangle with a 90° angle in it), I can use the

Pythagorean Theorem to calculate the exact length of the missing third side. I will not

attempt to use the Pythagorean Theorem unless I have evidence that it is a right triangle.

Target G4: Given the graph of a line with a definite y-intercept and slope, I can write the equation of

the line in slope-intercept form (y = mx + b). If the line is vertical, I can write its equation

in the form x = #.

Target G5: I can accurately graph a line given in slope-intercept form without a calculator.

Geometry 263 Prerequisites

Geometry 263 Prerequisites page 2

Target N1: I can multiply a fraction and a whole number without a calculator.

Examples:

A) 3

7(56) B) 1

2(14 9) C) 1

8(72+64)

1. Multiply the following, without a calculator.

A) 2

3(24) B) 5

428 C)

4

5(45)

D) 1

3(18 9) E) 3

4(12 7) F) 2

5(8 15)

G) 5

6(54+48) H)

1

9(63+45) I)

2

7(49+56)

Target N2: I can simplify the square root of a whole number without a calculator.

You undoubtedly know that 25 will simplify to 5, but what will 24 simplify to? In the

screen shot shown below you can see an exact answer and an approximate answer.

Numbers

Exact

Approximate

Geometry 263 Prerequisites page 3

When the square root of a number does not have a whole number answer, it may still be

possible to simplify it. First we need to review the following property of real numbers:

If a and b are both non-negative real numbers, then a b = a b .

To simplify a square root you first factor the number into primes and then (if possible) look

for any perfect square factors that you can take the square root of. The factorization of 24

into primes is 24 = 2∙2∙2∙3.

Using the property above we can write 24 = 4 6 = 4 6 . We do this is because we

know that 4 =2 and we leave 6 as it is. Therefore 24 can be simplified to 2 6 . Many

numbers e.g.: 6, 7, 10 do not have perfect square factors and cannot be simplified.

Examples: simplify the following square roots without a calculator.

A) 18 B) 75 C) 90 D) 300

2. Simplify (if possible) the following square roots without a calculator.

A) 99 B) 8 C) 45 D) 20

E) 50 F) 56 G) 63 H) 150

I) 200 J) 15 K) 80 L) 32

Geometry 263 Prerequisites page 4

Target N3: I can multiply square roots without a calculator.

To multiply square roots we need to review the following property of real numbers:

If a and b are both non-negative real numbers, then p a q b=p q a b .

The "outside" numbers multiply together, and the "inside" numbers multiply together.

Since it might be necessary to simplify the square root, you should not actually multiply the

a and b, but rather use them as the factor tree to see if there are any perfect squares.

Examples: multiply (and simplify) the following square roots without a calculator.

A) 1

248 B) 5 2 3 7 C) 5 2 3 6 D) 1

210 3 6

3. Multiply (and simplify) the following square roots without a calculator.

A) 1

318 B) 5 7 4 2 C) 10 6

D) 1

214 6 E) 3 15 10 F) 3 5 3 5

G) 2

3 5 H) 5 6 3 8 I) 5 2 3 8

Geometry 263 Prerequisites page 5

Target N4: I can add square roots without a calculator.

Square roots can be added only when the number under the square root is the same for

both terms: p qa a is the same as p q a( ) . Note that adding square roots is similar

to adding like terms: 5x + 4x = (5 + 4)x = 9x, and 5 7+4 7 =(5+4) 7=9 7 .

Examples: add (and simplify) the following square roots without a calculator.

A) 10 5 6 5 B) 5 3 6 5 C) 5 2 18 D) 14 3 – 75

4. Add (and simplify) the following square roots without a calculator.

A) 5 2+7 2 B) 30 3 –5 3 C) 5 + 6

D) 3 2 + 8 E) 20 + 80 F) 10+ 3

G) 12+2 27 H) 75+ 25 I) 1

210+3 10

Geometry 263 Prerequisites page 6

Target A1: I can multiply polynomials (monomials and binomials) without a calculator.

Examples:

A) x x3 2

9 4 B) 5x(4x – 7) C) (x + 6)(x – 2) D) x2

( +10)

5. Multiply the polynomials without a calculator.

A) x x2

7 8 B) (4x)(11x) C) x x3 1

216 D) x x x2

3(6 4 )

E) 3(4x – 9) F) -6(2x – 3) G) x x2

3 (4 –5) H) x x+1

3(12 9)

I) x x–23

5(15 20) J) (x + 5)(x + 10) K) (x – 3)(x – 7) L) (x + 9)(x – 4)

M) (2x + 3)(x + 5) N) x x1

2+5 4 –6 O) x

2( –3) P) x+

2(2 5)

Algebra

Geometry 263 Prerequisites page 7

Target A2: I can solve linear equations without a calculator.

Examples:

A) 4x + 10 = 7x – 5 B) x3

5+10=40 C) 4 – 2(x – 3) = 19

6. Solve each equation for x without a calculator.

A) 8x + 11 = 3x – 39 B) 3(2x + 1) = x + 38 C) 4 + 6(x + 8) = 16

D) 3x + 8 + 2(x – 1) = 36 E) 4 – (3x + 2) = 17 F) 16 – 2(x + 5) = 4

G) 5x – (2x + 8) = 19 H) x1

2–7=11 I) x1

3+5=20

J) x2

3+5=11 K) x3

4–9=15 L) x x4

3–1= +8

Geometry 263 Prerequisites page 8

Target A3: I can solve proportions, including those that lead to x n2= (where n is a whole number)

without a calculator.

Examples:

A) x

2 7=

3 +6 B)

x

x

4=

9 C)

x

x

2=

12

7. Solve each proportion for x without a calculator.

A) x 3

=8 2

B) x

2 3=

5 – 4 C)

x

4 5=

2 +7 9

D) x x

4 3=

–2 E)

x

x

5=

20 F)

x

x

10=

5

Target A4: I can solve a system of linear equations by the elimination method.

Examples:

A) x y

x y

3 + =17

2 – =8 B)

x y

x y

2 + =13

5 –4 =52 C)

x y

x+ y

2 +3 =15

5 1=-2

Geometry 263 Prerequisites page 9

8. Solve each system of linear equations (you may use a calculator for arithmetic).

A) x y

x y

2 – =10

4 + =26 B)

x y

x y

2 +3 =9

–5 =24 C)

x y

x y

2 + =1

5 +3 =6

D) x y

x y

4 +3 =32

3 – =24 E)

x y

y+ x

2 +6 =21

–4 =7 F)

y x

x y

3 – =38

5 +2 =-3

G) x y

x y

4 +3 =9

3 –2 =11 H)

x y

x y

4 +6 =10

6 +5 =3 I)

x y

x y

3 +2 =10

5 +7 =13

Geometry 263 Prerequisites page 10

Target A5: I can solve a quadratic equation with factoring and without a calculator.

Examples:

A) x2–9=0 B) x x

22 –18 =0 C) x x

2–9 +20=0

9. Solve each quadratic equation with factoring and without a calculator.

A) x2–25=0 B) x

2–36=0 C) x x

2–8 =0

D) x x2

4 +20 =0 E) x x2=3 F) x x

2+7 +10=0

G) x x2+14 +33=0 H) x x

2–9 +14=0 I) x x

2–8 –33=0

J) x x2+3 =28 K) x x

2+36=12 L) x x

2=13 –40

Geometry 263 Prerequisites page 11

Target A6: I can solve a quadratic equation with the quadratic formula.

If a quadratic equation is in the form 2

0ax +bx+c = , then there are two solutions to this

equation: b+ b ac

x =a

2- –4

2 and

b b acx =

a

2- – –4

2.

Examples: solve the quadratic equations with the quadratic formula.

A) x x2

3 +5 –2=0 B) x x2

5 +2=7

10. Solve each quadratic equation with the quadratic formula (you may use a calculator for arithmetic).

A) x x2

7 +5 –2=0 B) x x2

5 –11 +2=0

C) x x2

4 +13 +3=0 D) x x2+756=60

Geometry 263 Prerequisites page 12

Target G1: I can determine the coordinates of a point in the xy-coordinate plane.

11. Use the xy-plane to determine the coordinates of each point.

A( , ) B( , )

C( , ) D( , )

E( , ) F( , )

G( , ) H( , )

Target G2: Given two ordered pairs in the xy-plane, I can calculate the distance between the points,

the slope of the line connecting the points, and the coordinates of the midpoint.

For two points x y1 1

( , ) and x y2 2

( , )…

The distance between the points is x x y y2 2

1 2 1 2– + –

The slope of the line connecting the points is y y

x x

1 2

1 2

–

–

The coordinates of the midpoint is

x x y y1 2 1 2+ +

,2 2

Example:

x y1 1

(11,27) and yx 22

(35,17)

Distance:

Slope:

Midpoint:

Basic Geometry

Geometry 263 Prerequisites page 13

12. Determine the length of the segment, the slope of the segment, and the midpoint of the segment.

A)

Distance: Slope: Midpoint:

B)

Distance: Slope: Midpoint:

C)

Distance: Slope: Midpoint:

D)

Distance: Slope: Midpoint:

Geometry 263 Prerequisites page 14

E)

Distance: Slope: Midpoint:

F)

Distance: Slope: Midpoint:

Target G3: Given two side lengths of a right triangle (a triangle with a 90° angle in it), I can use the

Pythagorean Theorem to calculate the exact length of the missing third side. If the right

angle is missing, I will not attempt to use the Pythagorean Theorem.

In the right triangle shown at right, the Pythagorean Theorem states that

222

(leg #1) + leg #2 = hypotenuse

(leg #1 and leg #2 are interchangeable and

always intersect at the right angle)

Example: determine the length of the missing side.

13 cm

12 cm

leg #2

hypotenuse leg #1

Geometry 263 Prerequisites page 15

13. Use the Pythagorean Theorem (when appropriate) to determine the exact length of the missing side.

A) B) C)

D) E) F)

G) H) I)

53 cm

45 cm

12 mm

11 mm

20

ft

42.5 ft

10 km

10

.5 k

m 91 ft

109 ft

19.5" 26" 4' 7'

18 cm

22 cm

33

"

56"

Geometry 263 Prerequisites page 16

Target G4: Given the graph of a line with a definite y-intercept and slope, I can write the equation of

the line in slope-intercept form (y = mx + b). If the line is vertical, I can write its equation

in the form x = #.

Example: write the equation of the line shown at right in slope-intercept form

14. Write the equation of each line (use the slope-intercept form when possible).

A) B) C)

D) E) F)

Geometry 263 Prerequisites page 17

G) H) I)

Target G5: I can accurately graph a line given in slope-intercept form.

Example: accurately graph y x52=- +6 .

15. Graph each line. Include the y-intercept and at least two other integer valued coordinates on the line.

A) y = 3x – 2 B) y x14= +5 C) y x+2

3=- 1

D) y x45=- –3 E) y = -2x + 6 F) y = -3

Geometry 263 Prerequisites page 18

1. A) 16 B) 35 C) 36

D) 54 E) 63 F) 48

G) 85 H) 12 I) 30

2. A) 3 11 B) 2 2 C) 3 5 D) 2 5

E) 5 2 F) 2 14 G) 3 7 H) 5 6

I) 10 2 J) 15 K) 4 5 L) 4 2

3. A) 2 B) 20 14 C) 2 15

D) 21 E) 15 6 F) 45

G) 45 H) 60 3 I) 60

4. A) 12 2 B) 25 3 C) 5 + 6

D) 5 2 E) 6 5 F) 10+ 3

G) 8 3 H) 5 3+5 I) 7

210 =3.5 10

5. A) x3

56 B) x2

44 C) x4

8 D) x3

16

E) 12x – 27 F) -12x + 18 G) x x3 2

12 –15 H) x x2

4 +3

I) x x3 2

9 –12 J) x x2+15 +50 K) x x

2–10 +21 L) x x

2+5 –36

M) x x2

2 +13 +15 N) x x2

2 +17 –30 O) x x2–6 +9 P) x x

24 +20 +25

6. A) x = -10 B) x = 7 C) x = -6

D) x = 6 E) x = -5 F) x = 1

G) x = 9 H) x = 36 I) x = 45

J) x = 9 K) x = 32 L) x = 27

7. A) x = 12 B) x = 23

2= 11.5 C) x = 1

10= 0.1

D) x = -6 E) x = -10 or x = 10 F) x =-5 2 or x =5 2

Answers

Geometry 263 Prerequisites page 19

8. A) x = 6 and y = 2 B) x = 9 and y = -3 C) x = -3 and y = 7

D) x = 8 and y = 0 E) x = 9 and y = 1

2 F) x = -5 and y = 11

G) x = 3 and y = -1 H) x = -2 and y = 3 I) x = 4 and y = -1

9. A) x = -5 or x = 5 B) x = -6 or x = 6 C) x = 0 or x = 8

D) x = 0 or x = -5 E) x = 0 or x = 3 F) x = -2 or x = -5

G) x = -3 or x = -11 H) x = 2 or x = 7 I) x = -3 or x = 11

J) x = -7 or x = 4 K) x = 6 L) x = 5 or x = 8

10. A) x = -1 or x = 2

7 B) x = 1

5 or x = 2

C) x = -3 or x = 1

4- D) x = 18 or x = 42

11. A) (5, 3) B) (-2, 4)

C) (-5, 0) D) (1, -6)

E) (-3, -4) F) (0, -2)

G) (5, -3) H) (3, 0)

12. A) Distance: 14 Slope: 0 Midpoint: (10, 8)

B) Distance: 12 Slope: undefined Midpoint: (11, 13)

C) Distance: 15 Slope: 4

3- Midpoint: (8.5, 8)

D) Distance: 34 Slope: 8

15 Midpoint: (18, 13)

E) Distance: 29 Slope: 20

21 Midpoint: (40.5, -36)

F) Distance: 514 Slope: 17

15- Midpoint: (1.5, -4.5)

13. A) 28 cm B) 37.5 ft C) 65 in

D) 14.5 km E) Impossible to complete F) 60 ft

G) 32.5 in H) 23 mm I) 65 ft

14. A) y x32= –6 B) y x1

4=- –3 C) y x43= +8

D) y x54=- E) y = 7 F) y = -2x + 10

G) y = 5x – 15 H) x = -5 I) y x52= +13

Geometry 263 Prerequisites page 20

15. A) B) C)

D) E) F)