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PAP Geometry Unit 6 Packet 2014-2015
Essential Questions:
• What is the difference between similar and congruent figures?How are they the same?
•
How can I compare and contrast objects using geometricattributes? How does this help me in a proof?
•
How are properties of geometric figures related to their measurable attributes??
Website: www.crhspapgtgeometry.pbworks.com
TUTORIALSMondays 2:45 - 3:!5pm Myers Room 2208Tuesdays 6:45 - 7:15am Turner Room 2201Wednesdays 2:45 - 3:30pm Mu Alpha Theta Room 2215Thursdays 6:45 - 7:15am Bynum Room 2227Fridays By appointment
4th Six Weeks 2014-2015 Geometry PreAP
MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY January 5 6 7 8 9
Teacher Workday 6.1 Geometric Means Golden Ratio Activity HW: Finish Worksheets Make notecards
6.1 Geometric Means WS 6.1 C #3-39 mult. of 3;Challenge 6.1 #2-8ev;
6.2 Proportions WS 6.2 #2-14even Begin 6.1-6.2 WS
6.3 Similar Polygons HW: WS 6.3C #2-18 even; Begin WS 6.3-6.6
12 13 14 15 16 Quiz 6.1-6.3 Notecard Check Finish WS 6.1-6.2
6.4/6.5 AA, SSS, SAS Similarity
Discovery Activities on AA,SSS, SAS
HW: WS 6.4C #2-16even; WS 6.5C #2-18even; Cont.WS 6.3-6.6
6.4/6.5 AA, SSS, SAS Similarity
Discovery Activities on AA,SSS, SAS
HW: WS 6.4C #2-16even; WS 6.5C #2-18even; Cont.WS 6.3-6.6
6.6 Proportionality Theorems Construction p.401 #19 WS 6.6C #2-22 even; Finish WS 6.3-6.6
Quiz 6.4-6.6 Chapter 6 Review WS
19 20 21 22 23
No School
6.7 Dilations
HW: WS 9.7
6.7 Dilations
HW: WS6.7C #2-12 even; Challenge #1-5
Chapter 6 Test Review HW: p.422 # 1-13
Chapter 6 Test
HW: Algebra Practice
26 27 28 29 30 7.1 Pythagorean Theorem WS 7.1 C #3-27mult of 3 omit 18,21;, WS 7.1-
7.3 SectB #1-18
7.2 Pythagorean Theorem Converse WS 7.2C #3-21mult of 3
Challenge #2-12 even
7.3 Geometric Mean
Activity p.448 WS 7.1-7.3 Sect A # 1-247.2
7.3 Geometric Mean “Altitude on Hypotenuse” WS
WS 7.3 C # 2-14even
Quiz 7.1-3
WS 7.1-7.4(Sect. A &B)
2 3 4 5 6 7.4 Special Right Triangles
WS 7.1-7.4 (Sect.D)
7.4 Special Right Triangles
WS 7.4C #1-18all
Review 7.1-7.4
WS 7.1-7.4 (Sect. C)
Test Chapter 7 Sections 7.1-7.4
Cumulative Review Chapters 1-6 #1-35
7.5 Tangent Ratios Start Trig Ratios WS WS 7.5C #1-18,21,24,26
9 10 11 12 13 7.6 Sine & Cosine Ratios Angle of elevation & depression problems WS 7.6C #1-23 Continue Trig Ratios WS
7.7 Solve Right Triangles WS 7.7C #1-24
FinishTrig Ratios WS
Cumulative Test Applications Applications
16 17 18 19 20 Student Holiday Applications Quiz 7.5-7.7
Review WS “A”
Test Review
Review WS “B”
Early Release
Test Review
*ALL ASSIGNMENTS ARE SUBJECT TO LAST MINUTE CHANGES!!
6.1 Ratios, Proportions and the Geometric Mean
Ratio: Ira and b are two numbers or quantities, measured in the same units, and b :/: 0,
then the ratio of a to b can be written as or .
Equivalent ratios: Two ratios that have the same simplified form.
Example 1 Simplify each ratio.
a. 76cm' 8cm b. 4ft2 4 irt
Example 2: Extended ratios
The measures of the angles in AABC are in the extended ratio of 2:3:4. Find the measures
of the angles.
Proportion: Two equal ratios
C
b da and d are called the
b and c are called the
1
Cross Product Property of Proportions: In a proportion, the product of the extremes
equal the product of the means.
a _ c , b:ÿ0 and d:ÿ0, thenIf ÿ a
Example 3 Solve the proportions.
3 2X+8c-1 c h. -- =ÿa, ÿ ÿ4 6 x+5 8
3 9 6ye,-ÿÿ
x 2x+5 20
Example 4
CD's are on sale at a music store. During the sale, you can purchase 3 CD's for $27.99.
You want to purchase 5 of your favorite CD's. How much will pay for them?
2
Geometric Mean: The geometric mean of two positive numbers a and b is the positive
number x that satisfies a = xx b
SO X2 -" __ and x =.
Example 5
Find the geometric mean of 12 and 14.
3
"Golden Prediction"
1) Onfrom 1
2) Writeline 1.
3) Writeline 2.
a blank sheet of paper number vertically
to 25 in the margin.
any whole number less than 100 on
(i.e. no negatives, fractions, decimals,
another whole number less than 100
...)
on
4) Add your numbers from lines i and 2 and write
the sum on line 3.
5) Add lines 2 and 3 together and write the sum
on line 4.
6) Add lines 3 and 4 together and write on line 5.
7) Continue adding the sum to the previous
number until you
8) Divide line 25 by line
to 9 decimal places.
9) Calculate (1+ÿ/-5)/2.
have filled out all
24 and write
Write this
25 lines.
your answer
answer to 9
decimal places below your answer to step 8.
10) How close were your answers to 8 and 9?
4
How "Golden" are you?
°Make the following measurements (in centimeters) with a partner. Record your own
measurements below. Your parmer will keep his own measurements on his paper.o Measure the Height from top of head to bottom of feet. Call it B.o Measure the Naval height from the floor to the naval. Call this measure N.o Measure the length of an index finger and call it F.o Measure the distance form the big knuckle in the "middle" of the index finger to
the finger tip. Call it K.o Measure the length of a leg from hip joint to the floor and call it L.o Measure the distance from the hip joint to the knee of the same leg measured
above. Call it H.o Measure the length of an arm from shoulder to the fingertips and call it A.o Measure the distance from the elbow to the fingertips of the same arm above and
call it E.o Measure theo Measure theo Measure theo Measure the
distance from the top of the head to the chin and call it C.distance from the center of the eyes to the chin and call it Y.circumference of the head and call it M.circumference of the neck and call it I.
B= N= F=
K= L= H=
A= E= C=
y= M= I=
2. Calculate the following ratios. Round all answers to 3 decimal placesa) B/N=b) FiK =c) L/H =d) A/E =e) C/Y =I) M/I=
l+,/g _3. Use a calculator to determine the value of this expression. 2
4. Which of your answers in section 2 was the closest to this number?
5
6
7
8
6.2 Proportions to Solve Geometry Problems
Additional Properties of Proportions
Reciprocal Property: If two ratios are equal, then their reciprocals are also equal.
g c- = - thenIf b d'
Interchange Mean Property: If you interchange the means of a proportion, then you form another true
proportion.
12 ¢If g = a'then
Other Property: In a proportion, if you add the value of each ratio's denominator to its numerator, then
you form another true proportion.
12 CIf ÿ- = 3' then
Example 1
In the diagram AB AC= -- m'ite four true proportions.DE DE ÿ
B 9 E
12
A C8 F
9
SV su2 If = ÿ, find VT and ST.Exam leo VT UR
12
VU
6
R T
Scale Drawing: a drawing that is the same shape as the object it represents but is a different size.
Scale (scale factor): a ratio that describes how the dimensions in the drawing are related to the actual
dimensions of the object,
Example 3
The scale of a map is lin.: 1440ft, Find the actual length of a street if the distance on the map is 3 inches.
10
11
12
13
Worksheet on Sections 6.1 & 6.2
Do on own paper. In #1 – #10, true or false.
1. 713
is an example of a proportion.
2. The value of a ratio may be greater than 1.
3. The ratio 4 : 3 is the same as the ratio 3 : 4.
4. If 4 is added to the numerator and to the denominator of a fraction, then the new ratio is equal tothe given ratio.
5. If the numerator and the denominator of a fraction are multiplied by 5, then the new ratio is equalto the given ratio.
6. 4 : 3 = 20 : 12 is a true statement.
7. It is possible for the second and third terms of a proportion to be the same number.
8. If the ratio of the lengths of two segments measured in meters is 2 : 3, then the ratio of thelengths when measured in centimeters is less than 2 : 3.
9. The means of the proportion 5 : 6 = x : y are x and y.
10. The second term of ab
= xy
is x.
Solve for x. All answers must be exact.
11. 13
xx−+
= 55
xx−+
12. 2 47
x a− = 56
x a+ 13. 328
x =
123314
14. Find x & y: 8x
= 5663
= 189
y
Solve. Write a sentence answering the problem.
15. The ratio of the measures of two complementary angles is 3 : 4. Find the measure of each angle.
16. Find the measure of the acute angles of a right triangle if their measures are in the ratio of 7 to11.
17. Find the measures of two supplementary angles if their measures are in the ratio of 4 to 5.
18. Coach Crandall's summer basketball camp has a maximum player-coach ratio of 12 : 1. How manycoaches must Coach Crandall hire for 201 players?
14
19. ME | | IH . If MC = 16, EH = 4, EC = 12, find MI and IC.Explain your answer.
20. A copy machine can increase a figure by the ratio of 2 : 3. Find the dimensions of the figure below whenit is increased:
AC = ________ m∠A = __________
AB = ________ m∠B = __________
BC = ________ m∠C = __________
Simplify.
21. 2 feet3 yards
22. 2 meters120 cm.
23. 500 grams1.5 kg.
24. 24 oz.3 lbs.
Solve the proportion.
25. The recommended application for a particular type of lawn fertilizer is one 50 pound bag for 575 squarefeet. How many bags of this type of fertilizer would be required to fertilize 2850 square feet of lawn?
26. You have just moved into a new neighborhood and a new house valued at $110,000. If your next doorneighbor pays $1,150 in real estate taxes each year on a house valued at $89,000, how much a year shouldyou expect to pay in real estate taxes? (Assume the rate is the same.)
27. A tractor which runs two-cycle engine requires gasoline mixed with oil. The gasoline to oil ratio is 30 : 1.How much gasoline is required to produce a mixture that contains 1 pint of oil?
28. A quality control engineer for a certain buyer found that the ratio of defective units to total units is 1 : 35.At this rate, what is the expected number of defective unites in a shipment of 28,000?
29. The ratio of snowfall in January to total snowfall during a given winter is 7 : 10. If 84 inches fell inJanuary, find the total snowfall for the entire winter.
30. A car’s estimated miles per gallon is 32. How many gallons would be used for a trip of 496 miles?
36° 36
12 24
34° C
B
A
15
6.3 Similar Polygons
Similar ÿ: Two polygons are similar if their corresponding angles are congruent and the lengths ofcorresponding sides are proportional.
The symbol ~ means similar.
Scale Factor of Two Similar Polygons: If two polygons are similar, then the ratio of the lengths of twocorresponding sides is called the scale factor.
Example 1
A
AABC ÿ ADEF
B24
3C
D
3 E
6
F
a. List all pairs of congruent angles.
b. Check the ratios of the corresponding side lengths.
c, Write the ratios of the corresponding side lengths in a statement of proportionality,
d. What is the scale factor of AABC to ADEF?
16
Example 2: Find the value ofx ifAABC ~ AGHJ.
B Hx
12 8
A 18 C
16
G 12 J
Theorem: Perimeters of Similar Polygons.If two polygons are similar, then the ratios of their
corresponding
are equal to the ratio of their
R
SM
N
L P Q
If KLMN ,-, PQRS,
KL+LM+MN+NKthen --" ÿ ÿ---
PQ+QR+RS+SP
Corresponding lengths in similar oob,,gons: If 2 polygons are similar, then the ratio of any 2 correspondinglengths in the polygons is equal to the scale factor of the similar polygons.
17
*This includes altitudes, medians, angle bisectors and perpendicular bisectors
Example 3: AMNP ÿ zXRST Find the length of the altitude NL.
N
30 27
M L pR T
20 18
S
18
19
20
Lesson 6-4 Prove Triangles Similar by AA
.(NC #84) Anÿ:le-Anÿle Similarity Postulate: If 2angles of one triangle are congruent to 2 angles ofanother triangle, then the two triangles are
Examplel Determine whether the triangles are similar.
C
B D
A > E
A
BF
E20° 65
CD
B
it3
21
VEx.2) Complete the proof:
W
Given: WY ]l VZProve: A XWY ~ A XVZ X
Z Y
Statements1.)2.) lXWY -ÿ ÿXVZ3.) ZXYW-ÿ ZXZV4.)AxwY~ A XVZ
Reasons
1.)2.)3.)4.)
Ex.3) Find the coordinates for point E so thatzxABC --./,ADE. A(0,0) B(0,4) C(8,0) D(0,5) E(x,y)
22
AA Similarity Activity
Supplies: Ruler, protractor, pencil, white paper
I. Create triangle ABC by following these steps.1) Draw and label segment AB 80 mm in length on the top half of your paper.2) Use protractor to measure and draw a 40 degree angle above segment with A
as the vertex. This will become side AC of the triangle after the next step.3) Use protractor to create and draw a 60 degree angle with B as a vertex. As you
draw this angle it will intersect with the segment from step 2. Label theintersection point C.
II. Create triangle XYZ by following these steps.4) Draw and label segment XY 120 mm in length on the bottom half of your paper.5) Use protractor to measure and draw a 40 degree angle with X as the vertex.
This will become side XZ of the triangle after the next step.6) Use protractor to create and draw a 60 degree angle with Y as a vertex. As you
draw this angle it will intersect with the segment from step 2. Label theintersection point Z.
III. Measurements (use mm for segment lengths)7) Measure of angle C =8) Measure of angle Z =9) LengthIOf AC =10) Length of BC =11) Length of XZ =12) Length of YZ =13) XY/AB =14) YZ/BC =15) ×Z/AC =
as fraction and decimalas fraction and decimalas fraction and decimal
IV. EvaluationDo you think your triangles are similar? Why or why not? What is thescale factor?
23
24
25
6.5 Proving that Triangles are Similar by SSS and SAS
(NC#85) Side-Side-Side (SSS) Similarity Theorem:If corresponding of two triangles areproportional, then the two triangles are similar.
to anlengths of the
Side-Anÿle-Side(SAS) Similarity Theoremof one triangle is congruentof a second triangle and the
including theseangles are proportional, then the triangles aresimilar.
Ex. 1) Name the theorem or postulate that can be usedto prove that A FGH ÿA JLK. /_H and/_K are rightangles.
8
H
FJ
4
26
Ex.2) Use the SSSA ABC ...A YZW.
Similarity theorem to showFind the value of "x", BC, &YZ.
A 4 W
2O 0 0
B C2x+3 Z
Ex.3) In the figure below, Aand NM. o
OPN ÿ,A OLM. Find LM
1810
P
5
2O NY
L X M
27
Discovery Lesson on SAS and SSS Triangle Similarity
1. Draw a 3 cm. segment. Label it AB . Using a compass with a 6 cm. radius and draw a large arc using pt.A as the center. From pt. B draw a large arc using a 7 cm. radius. Label the intersection of the 2 arcs pt.C. Draw AC and BC . Using a protractor measure the angles.
AB = _________ m∠A =____________
BC = _________ m∠B =____________
AC = _________ m∠C = ___________
2. Now draw a 4.5 cm. segment. Label it DE . Using a 9 cm radius draw a large arc from pt. D. From pt. Edraw a large arc with a 10.5 cm. radius. Label the arcs intersection pt. F. Draw DF and EF . Measurethe angles.
DE = _________ m∠D = ___________
EF = _________ m∠E =____________
DF = _________ m∠F =____________
3. Are the triangles similar? Find the scale factor.
4. Construct a 75° angle. Label it ∠ABC. Make AB = 5 cm. and make BC = 7 cm. Draw AC . Measure allother lengths and angle measures.
AB = _________ m∠A =____________
BC = _________ m∠B =____________
AC = _________ m∠C = ___________
5. Now construct a 75°angle. Label it ∆XYZ. Make XY = 7.5 cm. and make YZ = 10.5 cm. Draw XZ .Measure all other lengths and angle measures.
XY = _________ m∠X =____________
YZ = _________ m∠Y = ____________
XZ = _________ m∠Z = ____________
6. Are the triangles similar? Find the scale factor.28
29
30
6.6 Use Proportionality Theorems
INC#87) Triangle Proportionality Theorem:,If a line to one side of a triangle
intersects the other sides, then it divides the two sides
a
R Sb
T > u
(NC#88) Triangle Proportionality Converse:If a line divides two sides of a triangle
, then it is parallel to the
X a
R
Y
S
tÿNC#89ÿ Theorem:two
If three intersect, then they divide the transversals
proportionally.
>>>
A
31
_Theorem:
triangle, then it divides thesegments whose lengths arelengths of the other two sides of the triangle.
If a ray bisects an angle of aintoto the
y Z
Ex. 1)o
proportions.
a.) WX=XY
Use the figures at the right to complete theW
X-Z is an angle bisector
b.) CB = EFEG
y Z
d.) CE=AG DG
B
Co.) GD=GF AB
32
Ex.2) Use the figure at the right to find the measure ofthe following segments.
a.) SQb.) QPc.) NRd.) OQ
M
Q
R
P
Ex. 3) Partition AB into 3 congruent segments using acompass and straight edge.
A B
33
34
35
Worksheet on Section 6.3 to 6.6
Do on your own paper. #1 – #6, true or false.
1. Every polygon is similar to itself.
2. Triangles are similar.
3. The ratio of the measures of two corresponding angles of two similar polygons is 1 : 1.
4. If the ratio of the lengths of two corresponding sides of two similar triangles is 1 : 1, then thetriangles are congruent.
5. If two triangles are congruent, then they are also similar.
6. If two triangles are similar, then they are congruent.
7. If ∆JUD ~ ∆BEK, find the following: (Hint: Factor 𝐸𝐸𝐸𝐸����and 𝐵𝐵𝐸𝐸���� first!)
BE = _________ EK = __________
UD = _________ BK = __________
JD = __________
8. Plot the following points: C(2, 3); F(1, 1); H(0, –1); E(6, –1); I(4, 1). Draw ∆CHE. Draw FI .a. Find the slope of FI and HE . What conclusion can you make?b. Use this conclusion to help you prove ∆CHE ~ ∆CFI.c. What is the scale factor of this similarity statement?
Proofs
9. Given: ∠DFE ≅ ∠EBCProve: ∆DEF ~ ∆CEB
10. Given: AB | | DEProve: ∆ABC ~ ∆EDC
B
E
K
J
U
D
x2 – 23x2 + 3x + 2
x2 + 4x + 3
7 x + 2
x + 3
36
Given: ∆RST with ∠RMN ≅ ∠S.
5. ∆RST ~ ∆______ Why?
6. Complete the extended proportion: ? ? ?RS ST RT
= = .
7. If RM = 4, MN = 5, and RS = 7, then ST = _____
8. If RM = 4, MN = 6, and ST = 8, then RS = _____
9. Given: BC ACDC EC
= 10. Given: ∠B and ∠D are rt. ∠s
Prove: ∆ABC ~ ∆EDC Prove: ∆ABC ~ ∆EDC
∆DEF with RS | | EF . 1. DR = 4; RE = 5; DS = 5; SF = 2. DS = 6; SF = 8; DR = 4; RE =
3. DE = 12; DR = 5; EF = 15; RS = 4. DR = 4; RE = 6; EF = 14; RS =
5. DR = x + 2; DS = 4x – 2; RE = 4x – 2; SF = 5x – 1; x =
6. RE = 4x – 6; SF = 6x – 5; DE = 2x + 6; DF = 8x – 2; DR =
7. Given: ∆DEF; ∠SRD ≅ ∠RFE R 8. Given: ∆DEF; FE DE⊥ ; RS DE⊥Prove: (DE)(RF) = (DF )(SE) Prove: (DR )(EF) = (DF )(RS)
9. Plot the following points: C(2, 3); F(1, 1); H(0, –1); E(6, –1); I(4, 1). Draw ∆CHE. Draw FI .a. Prove ∆CHE ~ ∆CFI using the SAS Similarity Theorem.b. Prove ∆CHE ~ ∆CFI using the SSS Similarity Theorem.
N
T S
M
R
F
D
S
E
R
S E D
F
37
11. Given: AC ⊥ DB ; ∠CDE ≅ ∠BAC
Prove: BAED
= ACDC
12. Given: DE is a midsegment of ΔABCProve: ΔABC ~ ΔDBE
13. Given: ΔABC is a right triangle. AD is an altitude.Prove: ΔABC ~ ΔDAC
14. Given: ∠ABC is a right angle. ∠EDC is a right angle.Prove: AB • DC = DE • BC
State whether the triangles are similar. Explain your answer. If so, state the scale factor.
1. 2.
3. 4.
3 .2
80°
30°
70° 80°
10 5
70°
30°
45º
45º
.2
15
32
12 50º
50º
38
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40
6.7 Identify and Perform Dilations
Dilation-
Center of dilation-
A dilation with center C and scale factor k is a transformation that maps every point P in the plane to a point P’, so that the following properties are true:
1.) If P is not point C, then CP’ is coincident to CP, and CP’ = k(CP), where k > 0 and k ≠ 1.
2.) If P is point C, then P = P’.
**The dilation is a _____________if 0 < k < 1 and it is an _______________ if k > 1.
41
**In a dilation, every image is similar to its preimage.
Coordinate notation for a dilation:
(x,y)→(kx,ky)
Example 1 Find the scale factor of the dilation. Then tell whether the dilation is a reduction or enlargement.
42
Scalar Multiplication-
Example 2 The vertices of XYZW are (-4,4); (2,8); (6,2); and (-2,-4). Use scalar multiplication to find the image of XYZW after a dilation with the center at the origin and a scale
factor of 12.
Example 3 The vertices of ∆LMN are L(2,-3);
M(3,-1)and N(4,-2). Find the image of ∆LMN after a translation (x,y)→(x-3,y+2) and dilation centered at the origin with scale factor 3.
43
Example 4- Perform a dilation with a scale factor of 2 with the center of dilation at point C.
C .
44
Worksheet 9.7
Find the coordinates of the image of the given point under a dilation with center O and given scale factor k.
1. A; k = 3 2. A; k =23 3. B; k = 0.5 4. B; k = 3
5. C; k =41 6. C; k = 1
Find the coordinates of the image of the given point under a dilation with center B and given scale factor k.
7. O; k = 4 8. O; k =21 9. A; k =
21 10. B; k = 0.5 11. C; k =
21
12. C; k = 1.5
C B
A O
45
46
47
48
Solve tile proportion.
5 _Y- 5I.6- 9 2.4 3,3-2b-3X 24 12 4 2
4, 7 _ 12a+8 a-I
In Exercises 5-7, use the diagram where APQR ÿ AABC.
5. List all pairs of congruent angles.
6. Write the ratios of the corresponding sides in a 12statement of proportionality.
7, Find the value of x, R
B 2O C
Determine whether the triangles are similar. If so, write a similaritystatement and the postulate or theorem that justifies your answer.
X 15 y 9. ÿ 10.
A E O
In Exercises 11-13, find the length of AB---.
11. B C D 12.o
A S
13,
A
3O8. N
2O
M J
B
3O
Determine whether the dilation from Figure A to Figure B is a reductionor an enlargement. Then find its scale factor.
15o "- '1 : i ! ; ;
i " FÿIÿ-: !ÿ
' = ' ' ' 9 Y:i:I-
i i i
16. SCALE MODEL You are making a scale model of yourschool's baseball diamond as part of an art project.The distance between two consecutive bases is
190 feet, If you use a scale factor of Tffÿ to build your
model, what will be the distance around the baseson your model?
422 Chapter 6 Similarity49
Geometry PreAP Name __________________ Algebra review Period ____
Date ________
I. Multiply or divide the monomials as indicated.
1. ( )( )2 3 2cd c d 2. ( )( )22 4e f ef 3. ( )26 p− 4. 22
3e
5. ( )( )534 2g h g− 6. ( )23 155r r− 7. 4
1x− 8. ( )76
10 5
c dc d−
9. 6
6
3yy
10. 10 5
11 5
102
a ba b
11. ( ) ( )( ) ( )
2 5
24 3
ab b
a ab12.
( )( )
14 2
4 3
x xy
x y
−
II. Multiply #13 – 17. Factor #18-24.
13. ( )24z + 14. ( )23 23t r− 15. ( )( )2 3 2 3x x− +
16. ( )( )3 2 3 22 7 2 7y z y z− + 17. ( )( )( )1 2 1x x x+ + − 18. 2 215 30cd c d+
19. 25 45a + 20. 2 2 5 6a b ab+ + 21. 23 2 16h h+ −
22. 26 7 2c c+ + 23. 2 215 16 4s st t− + 24. 232 18r r+ +
50
