Download ASM Geometry Summer Preparation Packet

The American School of Marrakesh
Geometry
Geometry Summer Preparation Packet
Summer 2016
Geometry Summer Preparation Packet
Thissummerpacketcontainsexcitingmathproblemsdesignedtoensureyourreadinessfor
Geometry.Thetopicscoveredinthispacketareconceptsthatshouldhavebeenmasteredin
coursesbeforeenteringGeometry.WhenyouenterGeometry,weassumeyouhavecertain
mathematicalskillsthatweretaughtinpreviousyears.Ifyoudonothavetheseskills,youwillfind
thatyouwillconsistentlygetproblemsincorrectnextyear,evenwhenyoufullyunderstandthe
geometryconcepts.Itcanbefrustratingforstudentswhentheyaretrippedupbythealgebraor
arithmeticandnotthegeometry.Thispacketisdesignedandintendedforyoutobrushupand
possiblyrelearntheseimportantconceptsnecessaryforsuccessinGeometry.
Showallworkthatleadsyoutoeachsolution.Useseparatepaper,ifnecessary.Youmayget
helpwiththispacketfromfriends,atutor,theinternet,orateacher,butpleaseunderstandthat
“help”meanshavingsomeoneexplainhowtosolvetheproblem,notjustsimplysupplyingthe
answerorcopyingtheworksomeoneelsedid.YOUareresponsibleforunderstandingthematerial
containedinthispacket,andforbeingabletoemploytheskillsnecessarytosolveeachproblem.
Allworkshouldbecompletedandreadytoturninonthefirstdayofschool.Thispacketwill
countaspartofyourfirsttermgradeandwillbegradedoncompletenessandcorrectness.Youwill
notbegivencreditforproblemsinthispacketifnoworkisshown.ASummerMathPackettestwill
begivenattheendofthefirstfullweekofschool.
DoalittleofyourSummerMathPacketeachweek.Youarenotexpectedtodoallofiton
thefirstdayorthelastweek.YourSummerMathPacketwillbeusedtoanalyzeyourstrengthsand
weaknesses,andassistyourteacherinhelpingyougrowmathematicallythroughouttheyear.
HonorandintegrityareattheheartofallstudentsatTheAmericanSchoolofMarrakesh.
TrueWarriorsnevercheat.Youareonlyhurtingyourselfbyattemptingtocopysomeoneelse’s
work.ThispacketistohelpyoubereadyforGeometry,andhelpyourteachersknowwhatyoucan
do.
Thankyouandhaveagreatsummer!
If you are receiving this packet, please provide the following information to me through
email to [email protected]
Name: ___________________________________________________________
Phone: ___________________________________________________________
Email: ___________________________________________________________
Emailing this information acknowledges receipt of the packet, and an understanding
that completion of this prerequisite packet is a requirement for the Geometry course.
The packet is due on the first day of class.
Introduction to Geometry 1
Name _________________________
SUMMER PACKET
for Algebra students entering Geometry
*Please complete every problem and SHOW ALL WORK. NO WORK = NO CREDIT. Write your answers on the
answer sheet at the end of the packet. This assignment will be graded for both accuracy and completion. You
will be tested during the second week of school on each of the following concepts.*
DETERMINING WHETHER A POINT IS ON A LINE
Example 1
Decide whether (3,-2) is a solution of the equation
( )
Substitute 3 for x and -2 for y.
Simplify.
The statement is true, so (3,2) is a solution of the equation
Exercises: Decide whether the given ordered pair is a solution of the equation.
(
1.
) ________
2.
(
3.
(
) _______
) ________
4.
(
5.
(
6.
(
) _________
) _________
) __________
CALCULATING SLOPE
Example 2
Find the slope of a line passing through (3,-9) and (2,-1).
Formula for slope
(
)
Substitute values and simplify.
Slope is -8.
Exercises: Find the slope of the line that contains the points
7. (4,1), (3, 6) _______
9.
8. (-8, 0), (5, -2)______
10. (0,-4), (7,3)______
Adopted from McDougal Littell Inc.Geometry
(5, 6), (9,8)_______
11. (-1, 7), (-3, 18)___
12. (-6, -4), (1, 10) ___
2 Introduction to Geometry
FINDING THE EQUATION OF A LINE
Example 3
Find an equation of the line that passes through the point (3, 4) and has a y-intercept of 5.
Write the slope-intercept form.
Substitute 5 for b, 3 for x, and 4 for y.
Subtract 5 from each side.
Divide each side by 3.
The slope is
. The equation of the line is
Exercises: Write the equation of the line that passes through the given point and has the given y-intercept.
13. (2, 1); b=5 _________
16. (7, 0); b=13______
14. (-5, 3); b=-12________
17. (-3, -3); b=-2_____
15. (-3, 10); b=8________
18. (-1, 4); b=-8_____
FINDING THE EQUATION OF A LINE
Example 4
Write an equation of the line that passes through the points (4, 8) and 3, 1). Find the slope of the line.
Substitute values.
Simplify.
( )
Substitute values into y = mx + b.
Multiply.
Solve for b.
Exercises: Write an equation of the line that passes through the given points.
19. (6, -3), (1, 2)___________
21. (5, -1), (4, -5)___________
23. (-3, -7), (0, 8)___________
20. (-7, 9), (-5, 3)___________
22. (-2, 4), (3, -6)__________
24. (1, 2), (-1, -4)___________
Adopted from McDougal Littell Inc.Geometry
Introduction to Geometry 3
Name _________________________
DISTANCE FORMULA
Example 5
Find the distance between
the points (-4, 3) and (-7, 8)
)
√(
√(
√(
(
)
(
))
)
(
)
( )
√
Exercises: Find the distance between the points
25. (3, 6), (0, -2)_________
27. (-3, 4), (1, 4)_________
29. (8, -2), (-3, -6)_______
26. (5, -2), (-6, 5)_________
28. (-6, -6), (-3, -2)_________
30. (-8, 5), (-1, 1)________
COMBINING LIKE TERMS
Example 6
Simplify
Group like terms
Simplify
Exercises: Simplify.
31. 6x + 11y – 4x + y
33. -3p – 4t – 5t – 2p
35. 3x2y – 5xy2 + 6x2y
32. -5m + 3q + 4m – q
34. 9x – 22y + 18x – 3y
36. 5x2 + 2xy – 7x2 + xy
Adopted from McDougal Littell Inc.Geometry
4 Introduction to Geometry
SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES
Example 7
Solve.
Subtract 5a from both sides
Add 12 to each side
Exercises: Solve the equation.
37. 3x + 5 = 2x + 11
38. 8m + 1 = 7m – 9
39. 11q – 6 = 3q + 8q
40. -14 + 3a = 10 – a
41. -2t + 10 = -t
42. -7x + 7 = 2x - 11
SOLVING INEQUALITIES
Example 8
Solve.
a.
b.
When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality
symbol to maintain a true statement.
a.
Exercises: Solve the inequality.
Adopted from McDougal Littell Inc.Geometry
b.
Introduction to Geometry 5
Name _________________________
WRITING AND SIMPLIFYING RATIOS
Example 9
a. Train A takes 35 minutes to travel its route. Train B, traveling the same route but making more stops, takes 47
minutes. What is the ratio of the time of Train A to Train B?
b. Jennie’s height is 4 feet, 7 inches. Her younger sister’s height is 25 inches. Find the ratio of Jennie’s height to her
sister’s.
Solutions
a.
(
b.
)
Exercises: Write the following ratios.
49.
Basmati rice needs to cook for 20 minutes, while quinoa (another grain) cooks for 25 minutes. What is the ratio
of cooking times for rice to quinoa?
50.
Jonathan caught 7 fish and Geogeanne caught 4. What is the ratio of fish caught of Jonathan to Georgeanne?
51.
Two sunflowers' growth was measured daily. At the end of the experiment, Sunflower A had grown from 2
inches to 2 feet, 3 inches. Sunflower B had grown from 3 inches to 2 feet, 6 inches. Find the ratio of the growth
in height of Sunflower A to Sunflower B.
Use the diagram at the right.
2
A
52.
What is the ratio of length to width of rectangle A?
53.
What is the ratio of the perimeter of rectangle A to the perimeter of rectangle B?
8
B
54.
What is the ratio of the area of rectangle A to the area of rectangle B?
4
DISTRIBUTIVE PROPERTY
Example 10
Solve.
a.
(
)
Adopted from McDougal Littell Inc.Geometry
b.
(
)
(
)
4
6 Introduction to Geometry
Exercises: Solve.
(
)
(
(
)
)
(
)
SOLVING PROPORTIONS
Example 11
Solve.
a.
b.
Exercises: Solve.
(
)
SIMPLIFYING RADICALS
Example 12
Simplify the expression √
√
√ √
√
Exercises: Simplify the expression.
√
√
√
√
√
√
√
√
√
√
√
√
Adopted from McDougal Littell Inc.Geometry
Introduction to Geometry 7
Name _________________________
SIMPLIFYING RADICAL EXPRESSIONS
Example 13
a.
√
√
b. ( √ )( √ )
√
√
√
√
c. ( √ )
√
√
√
Exercises: Simplify the radical expression.
√
√
√
√
√
( √
)(√
√
√
(√
)(√
( √ )( √ )
√
√
√
(√
)(√ )
√
√
(√
)(√
)
( √ )
)
( √ )
( √ )
(
)
√
√
SIMPLIFYING QUOTIENTS WITH RADICALS
Example 14
Simplify the quotient
√
√
√
√
√
√
√ √
√
Exercises: Simplify the quotient.
√
√
√
√
Adopted from McDougal Littell Inc.Geometry
√
√
√
√
√
√
√
√
√
√
√
√
√
)
8 Introduction to Geometry
SOLVING LITERAL EQUATIONS
Example 16
Given the formula for the surface area of a right cylinder, solve for h.
(
)
or
(
)
Exercises: Solve the literal equation for the indicated variable. Assume variables are positive.
(
)
ALGEBRAIC EXPRESSIONS
Example 17
a. Write an expression for seven less than a
number
b. Write an equation for three times less than six times a
number is five times the same number plus 5, then
solve.
Exercises: Write the expression or equation. Solve the equations.
114.
Half of a number plus three times the number
115.
The product of five and a number decreased by seven equals thirteen.
116.
Sixteen less than twice a number is 10.
117.
Twice a number increased by the product of the number and fourteen results in forty-eight.
118.
Half of a number is three times the sum of the number and five.
Adopted from McDougal Littell Inc.Geometry
Introduction to Geometry 9
Name _________________________
PERCENT PROBLEMS
Example 18
a. What number is 12% of 75?
( )
b. 6 is what percent of 40?
Exercises:
119.
What number is 30% of 120?
120.
11 dogs is what percent of 50 dogs?
121.
What distance is 15% of 340 miles?
122.
200 is what percent of 50?
123.
34 is what percent of 136?
124.
8 weeks is what percent of a year?
SIMPLIFYING RATIONAL EXPRESSIONS
Example 19
Simplify.
a.
=
(
(
Exercises: Simplify.
(
(
)
)
Adopted from McDougal Littell Inc.Geometry
)
)
=
b.
(
=(
)(
)(
)
)
=
001-018 SGWB 869623
11/30/04
3:04 PM
Page 15
NAME ______________________________________DATE __________PERIOD______
3-4
Study Guide
Student Edition
Pages 110–115
Adjacent Angles and Linear Pairs of Angles
Pairs of Angles
Special Name
Definition
Examples
adjacent angles
angles in the same plane that
have a common vertex and a
common side, but no common
interior points
!3 and !4 are adjacent angles.
adjacent angles whose
noncommon sides are opposite
rays
linear pair
5
6
!5 and !6 form a linear pair.
m!1 ! 45, m!2 ! 135, m!3 ! 125, m!4 ! 45, m!5 ! 135,
m!6 ! 35, and !CAT is a right angle. Determine whether
each statement is true or false.
1. !1 and !2 form a linear pair.
2. !4 and !5 form a linear pair.
6
1
3
2
3. !6 and !3 are adjacent angles.
4. !7 and !8 are adjacent angles.
5. !CAT and !7 are adjacent angles.
4
5
Use the terms adjacent angles, linear pair, or neither to describe
angles 1 and 2 in as many ways as possible.
6.
7.
8.
1
1
2
1
©
2
2
Glencoe/McGraw-Hill
15
Geometry: Concepts and Applications
001-018 SGWB 869623
11/30/04
3-5
3:04 PM
Page 16
NAME ______________________________________DATE __________PERIOD______
Study Guide
Student Edition
Pages 116–121
Complementary and Supplementary Angles
The table identifies several different types of angles that occur in pairs.
Pairs of Angles
Special Name
Definition
Examples
complementary
angles
two angles whose measures have
a sum of 90
supplementary
angles
two angles whose measures have
a sum of 180
30
60
160
20
Each pair of angles is either complementary or
supplementary. Find the value of x in each figure.
1.
2.
x
x
20
3.
x
65
15
4.
5.
6.
7. If m!P ! 28, !R and !P are
supplementary, !T and !P are
complementary, and !Z and !T
complementary, find m!R,
m!T, and m!Z.
©
Glencoe/McGraw-Hill
8. If !S and !G are supplementary,
m!S ! 6x " 10, and m!G !
15x " 23, find x and the measure
of each angle.
16
Geometry: Concepts and Applications
001-018 SGWB 869623
11/30/04
3:04 PM
3-6
Page 17
NAME ______________________________________DATE __________PERIOD______
Study Guide
Student Edition
Pages 122–127
Congruent Angles
Opposite angles formed by intersecting lines are
called vertical angles. Vertical angles are always
congruent. !1 and !3, and !2 and !4 are pairs
of vertical angles.
2
3
1
4
Identify each pair of angles in Exercises 1–4 as adjacent, vertical,
complementary, supplementary, and/or as a linear pair.
1. !1 and !2
2. !1 and !4
2
3. !3 and !4
1
4. !1 and !5
5. Find x, y, and z.
3
4
5
6. Find x and y if !CBD ! !FDG.
x
51
y
z
Use the figure shown to find each of the following.
7. x
8. m!LAT
!
!
9. m!TAO
10. m!PAO
©
Glencoe/McGraw-Hill
17
Geometry: Concepts and Applications
10 Introduction to Geometry
ANSWER SHEET
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Adopted from McDougal Littell Inc.Geometry
Introduction to Geometry
Name _________________________
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Adopted from McDougal Littell Inc.Geometry