Relationships of Side Lengths and Angle Measures. Theorem: Practice: State the longest side of the triangle. Justify your answer.
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1 10R Unit 8 Similarity CW 8.1 Relationships of Side Lengths and Angle Measures HW: Worksheet #8.1 (following these notes) Theorem: Practice: 1. State the longest side of the triangle. Justify your answer.. The ratio of A: B: C = 8:7:3. State the longest side of the triangle. Justify your answer. 3. State the longest side of the triangle. Justify your answer. 4. State the smallest angle of the triangle. Justify your answer. 5. Express the relationship of the angles of the triangle as an inequality in order from smallest to largest. Revised: 1/1/010 9:10 PM 1
2 6. State the longest side of the triangle. Justify your answer. 7. In triangle PQR, P: Q: R= 3::1. State the shortest side of the triangle and justify your answer. 8. In right triangle ABC, BC is the hypotenuse, AB = 9, and AC = 1. State the smallest angle of the triangle and justify your answer. 9. In right triangle PQR, PR is the hypotenuse, QR = 4, and PR = 5. State the smallest angle of the triangle and justify your answer. 10. In Δ ABC, side AC is extended through point D such that BCD is an exterior angle. If m A= 13x, m B= 10, and m BCD = x State the longest side of the triangle and justify your answer. 11. The ratio of the lengths of the sides of ΔABC is AB : BC : CA = 4:5:6. If the perimeter of the triangle is 75, state the largest angle of the triangle and justify your answer. Revised: 1/1/010 9:10 PM
3 10R Unit 8 Similarity HW 8.1 Relationships of Side Lengths and Angle Measures 1. Using the graph below, express the angles of the triangle as an inequality from least to greatest. HINT: You MUST use the distance formula three times.. Express the relationship of the angles of the triangle as an inequality in order from smallest to largest. 3. The perimeter of ΔABC is 78. Use the diagram to the right to determine the smallest angle of the triangle and justify your answer. 4. In obtuse ΔABC with obtuse C, side AC is extended through point D such that BCD is an acute exterior angle. If m A= 5x+, m B= 3x 4, and m BCD = x + 1. State the shortest side of the triangle and justify your answer. Revised: 1/1/010 9:10 PM 3
4 10R Unit 8 Similarity CW 8. Introduction to Similar Figures & Geometric Mean HW: Worksheet #8. (following these notes) Warm-up: Factor: x 15x+ 6 FOIL to check your answer: What is a ratio? Expressed 3 Ways: a.) b.) c.) The product of the equals the product of the. This is the new and improved way of saying:. Solving Proportions: Solve for the value(s) of x. Use ( ), you may have to distribute or FOIL. 1. a) 3 = b) 8 z + = m+ 5 m+ 1 z 4. In the diagram at the right, a b = 3. Complete each statement. 4 a) b a = b) 4a = c) b 4 = d) 7 4 = Symbol for Similar: Extended Proportions: Used to write a similarity statement. Revised: 1/1/010 9:10 PM 4
5 3. Given that ΔMNP ~ ΔSRT, state the corresponding angles that are congruent and write a similarity statement. M N P Similarity Statement: = = Scale Factor: Think Dilation 4. Using the diagram to the right: a) What type of triangle is illustrated? Explain. b) Find the scale factor of (image) Δ ABC to (pre-image) Δ DEF. This is also known as the ratio of sides. c) What is the ratio of the perimeters? d) What is the ratio of areas? If the ratio of sides is a:b, then the ratio of perimeters is : and the ratio of areas is :. Basic Similarity Word Problems: 5. Triangles ABC and HIJ are similar. The length of the sides of ABC are 10, 16, and 96. The length of the smallest side of HIJ is 3, what is the length of the longest side of HIJ? 6. Triangles HIJ and MNO are similar. The perimeter of smaller triangle HIJ is 44. The lengths of two corresponding sides on the triangles are 13 and 6. What is the perimeter of MNO? 7. Triangles GHI and MNO are similar. GI:MO = 4:3, and MN = 87, what is the length of GH? Revised: 1/1/010 9:10 PM 5
6 8. A tree 4 feet tall casts a shadow 10 feet long. Paul is 3 1/ feet tall. How long is Paul's shadow? 9. Triangles KLM and STU are similar. The length of the sides of KLM are x + 8, 4x - 6, and 5x The perimeter of KLM is 306. The perimeter of STU is 459, what is the length of the longest side of STU? Mean Proportional: is also known as the Geometric Mean To find the mean proportional between two given numbers: a.) b.) c.) Practice: Find the geometric mean of the two given numbers in simplest radical form. 10. and and and and 16 Revised: 1/1/010 9:10 PM 6
7 10R Unit 8 Similarity HW 8. Introduction to Similar Figures & Geometric Mean 1. Given that ABCD ~ EFGD: a) A C F b) Write a similarity statement: = = = c) What is the scale factor of ABCD to EFGD? d) Is this dilation an enlargement or a reduction? e) What is the value of x? What is the value of y?. Find the geometric mean of the two given numbers in simplest radical form. a.) 10 and 1 b.) 9 and Determine whether the polygons are similar. If so, write a similarity statement and give the scale factor. If not, explain. a) b) 4. A company produces a standard-size U.S. flag that is 3 ft. by 5 ft. The company also produces a giantsize flag that is similar to the standard-size flag. If the shorter side of the giant-size flag is 36ft, what is the length of its longer side? 5. Two polygons have corresponding side lengths that are proportional. Can you conclude that the polygons are similar? Justify your answer. 6. A cartographer is making a map of Pennsylvania. She uses the scale 1 in = 10 mi. The actual distance between Harrisburg and Philadelphia is about 95 mi. How far apart should she place the two cities on the map? 7. In the diagram to the right, ΔDFG ~ ΔHKM. Find each of the following. a) The scale factor of ΔHKM to ΔDFG. b) m K c) MK Revised: 1/1/010 9:10 PM 7
8 10R Unit 8 Similarity CW 8.3 Similarity Theorems HW: Worksheet #8.3 (following these notes) Warm-up: Simplify: x y 96x 150a Theorem: Two Methods: *Depends on the picture you are given! a a+ c 1. Top of the Mountain =. Side to Side b d a c = a+ b c+ d a b c = d Use when you have the bottom of the mountain. Use when you have the sides of the mountain. Practice: Solve for x Simplest radical form. 6. Simplest radical form. Revised: 1/1/010 9:10 PM 8
9 7. Determine whether PS QR. 8. Is BD CE? 9. In the diagram below, AB CD. Find the value of AC. What is another name for, in this case? 10.In Δ ABC, point D lies on AB and point E lies on BC such that DE AC. If AD = x, DB = 4, DE = x, and AC = 8, find AB. 11. In Δ PQR, point S lies on QR and point T lies on PR such that ST PQ. If RS = 3, PT = 0, RT = x, QS = 5, find PR in simplest radical form. 1. For each part, you are given a.) AB AE =. Find BC. BC ED b.) Revised: 1/1/010 9:10 PM 9
10 10R Unit 8 Similarity HW 8.3 Similarity Theorems Find each variable: 1) ) 3) 4) 5) 6) 7) 8) 9) In VLMN, point O lies on LM and point P lies on NM such that LN P OP. If LO = 10, OM = x, NM = 45, and PN = 30, find OM. 10) In ΔABC, point D lies on AB and point E lies on BC such that DE AC. If AD = 1, DB = 9, BE = x + 4 and EC = x, find BC. 1Revised: 1/1/010 9:10 PM 0
11 10R Unit 8 Similarity CW 8.4 Similarity Theorems - Practice HW: Worksheet #8.4 (following these notes) Warm-up: Solve for the value(s) of x: x 50= x + 3x+ 4 Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally (& vice versa). More Practice: Given AB FE =. Find AC. BC ED 1Revised: 1/1/010 9:10 PM 1
12 Two Other Similarity Theorems: If three parallel lines intersect two transversals, then they divide the transversals proportionally. If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. 7. Given a b c, solve for x. 8. Find the length of AB st Street, 4nd Street and 43rd Street 10. AD bisects CAB, find AC. all lie parallel. Find the distance between 4nd and 43rd running along Broadway. 11. Find the length of AB. 1. Solve for x. 1Revised: 1/1/010 9:10 PM
13 10R Unit 8 Similarity HW 8.4 Similarity Theorems - Practice 1. The ratio of the perimeters of two similar figures is 5 : 9. What is the ratio of the areas?. Using the extended ratios indicated in the drawing to the right, an artist can sketch the proper placement of the eyes, nose and mouth of an adult or infant s face. a) If AE = 7 cm in the diagram, find AB, BC, CD and DE. b) If PT = 1 inches, how far from the top should you place the line for the eyes? 3. Three campsites are shown in the diagram to the right. What is the length of Site A along the river? 4. Solve for x. a) b) c) d) 1Revised: 1/1/010 9:10 PM 3
14 e) f) 5. Solve for x. a) b) 6. In Washington DC, E. Capitol Street, Independence Avenue, C Street and D Street are parallel streets that intersect Kentucky Avenue and 1 th Street. a) How long (to the nearest foot) is Kentucky Avenue between C Street and D Street? a) How long (to the nearest foot) is Kentucky Avenue between E. Capitol Street and Independence Avenue? 1Revised: 1/1/010 9:10 PM 4
15 10R Unit 8 Similarity CW 8.5 HW: Finish this CW 8.5 Similarity Review Answers & Work = Consult Teacher Website Similarity Theorems: THEOREM: DIAGRAM: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If a line divides two sides of a triangle proportionally, then it is parallel to the third side. If three parallel lines intersect two transversals, then they divide the transversals proportionally. If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Can prove triangles similar by: AA need angles in each triangle (most common) SAS need 1 angle in each and verify 1 proportion SSS verify proportions 1Revised: 1/1/010 9:10 PM 5
16 1. A triangle has sides whose lengths are 5, 1, and 13. A similar triangle could have sides with lengths of (1) 3, 4, and 5 (3) 7, 4, and 5 () 6, 8, and 10 (4) 10, 4, and 6. The accompanying diagram shows two similar triangles. Which proportion could be used to solve for x? x (1) = () 4 15 x = (3) = (4) = 9 x 1 x In the accompanying diagram, ΔQRS is similar to ΔLMN, RQ = 30, QS = 1, SR = 7, and LN = 7. What is the length of ML? 5. In the accompanying diagram, triangle A is similar to triangle B. Find the value of n. 6. The Rivera family bought a new tent for camping. Their old tent had equal sides of 10 feet and a floor width of 15 feet, as shown in the accompanying diagram. If the new tent is similar in shape to the old tent and has equal sides of 16 feet, how wide is the floor of the new tent? 1Revised: 1/1/010 9:10 PM 6
17 7. In obtuse ΔPQR with obtuse Q, side RQis extended through point S such that PQS is an acute exterior angle. If m P= 6x+ 3, m R= x+ 6, and m SQP= 10x 5, state the shortest side of the triangle and justify your answer. 8. Triangle RST has a perimeter of 9. If RS = 5x + 1, ST = 3x + 5, and RT = 3x, which of the following is true? (1) S > T () S > R (3) T > S (4) T = R 9. Fran s favorite photograph has a length of 6 inches and a width of 4 inches. She wants to have it made into a poster with dimensions that are similar to those of the photograph. She determined that the poster should have a length of 4 inches. How many inches wide will the poster be? 10. Fran has another photograph that has a length of 5 inches. She had it increased by a scale factor of 4, so the ratio between them is 5:0 or 1:4. What is the ratio of the altitudes of the photos? What is the ratio of the perimeters? What is ratio of the areas? 11. Delroy s sailboat has two sails that are similar triangles. The larger sail has sides of 10 feet, 4 feet, and 6 feet. If the shortest side of the smaller sail measures 6 feet, what is the perimeter of the smaller sail? (1) 15 ft () 60 ft (3) 36 ft (4) 100 ft 1Revised: 1/1/010 9:10 PM 7
18 1. Two triangles are similar. The lengths of the sides of the smaller triangle are 3, 5, and 6, and the length of the longest side of the larger triangle is 18. What is the perimeter of the larger triangle? (1) 14 () 4 (3) 18 (4) The base of an isosceles triangle is 5 and its perimeter is 11. The base of a similar isosceles triangle is 10. What is the perimeter of the larger triangle? (1) 15 () (3) 1 (4) Pentagon ABCDE is similar to pentagon FGHIJ. The lengths of the sides of ABCDE are 8, 9, 10, 11, and 1. If the length of the longest side of pentagon FGHIJ is 18, what is the perimeter of pentagon FGHIJ? (1) 50 () 75 (3) 56 (4) Which is not a property of all similar triangles? (1) The corresponding angles are congruent. () The corresponding sides are congruent. (3) The perimeters are in the same ratio as the corresponding sides. (4) The altitudes are in the same ratio as the corresponding sides. 16. Two triangles are similar, and the ratio of each pair of corresponding sides is :1. Which statement regarding the two triangles is not true? (1) Their areas have a ratio of 4 : 1. () Their altitudes have a ratio of : 1. (3) Their perimeters have a ratio of : 1. (4) Their corresponding angles have a ratio of : To the nearest tenth, 1Revised: 1/1/010 9:10 PM 8
19 19. In the diagram to the right, lines a, b, and c are cut by transversals d and e such that a b c. Solve for x. 0. In the diagram to the right, AD bisects Solve for the value of x. BAC. Review Questions to Study: 1. In triangle ABC, AF FC, BD AC, and BE bisects ABC. Which of the following is false? (1) ABE CBE () BDis the altitude to AC (3) ABD CBD (4) BF is the median to AC. Which of the following is the correct equation of the circle in the graph to the right? (1) ( x ) ( y ) () ( x ) ( y ) (3) ( x ) ( y ) (4) ( x ) ( y ) = = = = 9 3a) Complete the following logic rules: Conjunction Disjunction Conditional Bi-Conditional p q p q p q p q T T T T T F T F F T F T F F F F b) Write the justification used for each truth table. A conjunction is only A conditional is only 1Revised: 1/1/010 9:10 PM 9
20 A disjunction is only A bi-conditional is only Sketch a diagram for each of the following: 4. Two planes perpendicular to the same line are (1) intersecting () perpendicular (3) skew (4) parallel 5. If two different lines are perpendicular to the same plane, they are (1) coplanar () consecutive (3) collinear (4) congruent 6. If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane (1) is parallel to the given plane () is perpendicular to the given plane (3) lies in the given plane (4) is parallel to the given line 7. When do you use the distance formula? Keywords: When do you use the midpoint formula? Keywords: When do you use the slope formula? Keywords: 8. a) Graph and label triangle ABC with A ( 1, 6), B (,1), ( 7,) b) Show ΔABC is an isosceles triangle. C. c) Find the coordinates of D, the point that bisects AC. d) Show AC BD Revised: 1/1/010 9:10 PM 0
21 10R Unit 8 Similarity CW 8.6 Similarity Proofs by Angle-Angle (AA~thm) HW: Worksheet #8.6 (following these notes) Warm-up: Write the equation of the line parallel to the line 4y+ x= 8 through the point (-, 10). To Prove Triangles are Similar: Symbol for Similar: 1. Given: DE AC Prove: a.) ΔDBE ~ Δ ABC b.) DB = DE AB AC c.) DB AC = AB DE 1. DE Statement AC. BDE BAC 3. B B 4. ΔDBE ~ Δ ABC Reason 5. DB DE = AB AC 6. DB AC = AB DE. Given: AB DE Prove: a.) ΔABC ~ Δ EDC AC AB b.) = EC ED c.) AC ED = EC AB 1. AB Statement DE. ACB ECD 3. B D 4. ΔABC ~ Δ EDC 5. AC AB = EC ED 6. AC ED = EC AB Reason Revised: 1/1/010 9:10 PM 1
22 What did the first two examples had in common, besides starting with Given? 3. Given: PS ST, PQ QR Prove: PS PR = PQ PT 4. Given: PQR TSR, PR QS Prove: ΔPQR ~ Δ TSR 5. Given: BA is the altitude to AC EDis the altitude to DF 3 4 Prove: AB BC = ED EF Revised: 1/1/010 9:10 PM
23 10R Unit 8 Similarity HW 8.6 Similarity Proofs by Angle-Angle (AA~thm) Complete the following. Proofs are nothing more than time and effort. 1. Given: BE CD Prove: AB CD = AC BE Statement 1. BE CD. ABE ACD 3. A A 4. ΔABE ~ Δ ACD Reason 5. AB BE = AC CD 6. AB CD = AC BE. Given: PQ TS Prove: ΔPQR ~ Δ TSR 3. Given: LM is the altitude to MP LN is the altitude to NO Prove: LM NO = MP LN Revised: 1/1/010 9:10 PM 3
24 10R Unit 8 Similarity CW 8.7 Similarity Proofs by SAS~thm & SSS~thm HW: Worksheet #8.7 (following these notes) Warm-up: Q is the incenter of Δ ABC. If QC = 50, FC = 48, and EQ = 7x, solve for the value of x. In addition to proving triangles similar by, you can also prove them similar by &. is the most common. To prove by, you need to verify proportion and have one pair of angles. To prove by, you need to verify proportions. Practice: Determine whether the following can be proven similar using the AA, SAS, SSS ~thm Revised: 1/1/010 9:10 PM 4
25 Extra Practice: 7. Given: ΔQRS is isosceles with vertex R QTU QRS Prove: QT RS = QR TU 8. Given: CA is the altitude to AB CDis the altitude to DE Prove: ΔABC ~ Δ DEC 9. Given: AB DE Prove: AB EC = DE BC Revised: 1/1/010 9:10 PM 5
26 Optional: SAS and SSS Proofs 10. Given: CD CA CE = CB Prove: ΔCDE ~ Δ CAB 11. Given: AB DE BC =, EF AC BC = DF EF Prove: ΔABC ~ Δ DEF 1. Given: AC DC = BC EC Prove: ΔABC ~ Δ DEC Revised: 1/1/010 9:10 PM 6
27 10R Unit 8 Similarity HW 8.7 Similarity Proofs by SAS~thm & SSS~thm Remember: Corresponding of triangles are. And: The product of the equals the of the. How many proportions do you need to verify for SAS~thm? How many proportions do you need to verify for SSS~thm? How many proportions do you need to verify for AA~thm? Practice: Determine whether the following can be proven similar using AA, SAS, SSS Revised: 1/1/010 9:10 PM 7
28 9. Given triangle ABC with A ( 1,1), B ( 4,1), C ( 1, 5) and triangle DEF with D(, ), E( 8, ), (, 10) prove that ΔABC ~ Δ DEF. F, What is the ratio of the perimeters of ABC to DEF? What is the ratio of the areas of ABC to DEF? 10. Given: AD CB Prove: CB DE = DA CE 11. Given: AE EF, AB BC Prove: a.) ΔAEF ~ Δ ABC AE EF b.) = AB BC c.) AB EF = AE BC Revised: 1/1/010 9:10 PM 8
29 10R Unit 8 Similarity CW 8.8 Review Packet HW: Finish this Review Packet! Answers = Consult Teacher Website Similarity Theorems: THEOREM: DIAGRAM: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If a line divides two sides of a triangle proportionally, then it is parallel to the third side. If three parallel lines intersect two transversals, then they divide the transversals proportionally. If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Can prove triangles similar by: AA need angles in each triangle (most common) SAS need 1 angle in each and verify 1 proportion SSS verify proportions Revised: 1/1/010 9:10 PM 9
30 Practice: 1. In the accompanying diagram of ABC, DE AB, CA = 9, DA = 3, and CE = 10. Find EB.. Triangle ABC is similar to triangle DEF. If AB = 6, BC = 7, DE = 1, and DF = 10, find the perimeter of ΔDEF and state the scale factor. 3. Triangle ABC is similar to triangle DEF. If AB = 4, BC = 6, DE = 6, and DF = 1, find the perimeter of ΔDEF and state the scale factor. 4. Given Δ PQR, S lies on PQand T lies on QR such that ST PR but ST is NOT a midsegment of Δ PQR, find the measure of ST if PS = 6, SQ = 3, and PR = For each of the following, find the length of AB. 6. The perimeter of a rectangular garden is 0 meters. The ratio of its length to width is 8 : 3. What is the length of the field? 3Revised: 1/1/010 9:10 PM 0
31 7. Find the length of BC. 8. Verify that ΔABC ~ ΔDEF for the given information below: Δ ABC : AC = 6, AB = 9, BC = 1 Δ DEF : DF =, DE = 3, EF = 4 How many proportions must be verified for SSS similarity? For SAS? 9. Triangles BCD and UVW are similar where BCD is the smaller triangle. The lengths of two corresponding sides on the triangles are 31 and 186. One side of UVW is 19. What is the length of the corresponding side on BCD? 10. Show that the triangles below are similar and write a similarity statement and state the scale factor. 11. Triangles CDE and IJK are similar. CE:IK = :6, and JK = 78, what is the length of DE? 1. Is ΔAEB ~ Δ ADC? If so, by what theorem? 13. If the ratio of the perimeters of two similar triangles is 4:5, what is the ratio of their areas? 14. If the ratio of the perimeters of two similar triangles is 8:3, what is the ratio of their areas? 3Revised: 1/1/010 9:10 PM 1
32 15. Determine whether the triangles are similar. If so, write a similarity statement and the postulate or theorem that justifies your answer. a.) b.) c.) 16. The measures of the angles of a triangle are in the extended ratio :3:5. Find the measures of the angles. 17. A tree casts a shadow that is 30 feet long. At the same time, a person standing nearby who is five feet two inches tall casts a shadow that is 50 inches long. How tall is the tree, to the nearest foot? **Watch the units! 18. The length of a rectangle is 0 meters and the width is 15 meters. Find the ratio of width to length of the rectangle. Then, simplify the ratio. 19. Find the geometric mean of each of the following in simplest radical form. a.) 8 and 1 b.) 3 and 15 c.) 7 and 14 d.) 3 and 6 0. In the diagram, BA = BC. Find BD. DA EC 3Revised: 1/1/010 9:10 PM
33 1. If AD = 1, BD = x, AE = x, and EC = x +. Find DB.. If AD = x, ED = 5, DB =, and BC = x +. Find AD. 3. Find the length of WT. 4. Two similar posters have a scale factor of 4:5. The large poster s perimeter is 85 inches. Find the small poster s perimeter. 5. Determine whether the polygons are similar. If so, write a similarity statement and find the scale factor. a.) b.) 6.The slope of a ramp is 3. If the rise of a similar ramp is 18, what is its run? 5 3Revised: 1/1/010 9:10 PM 3
34 7. Use the given information to determine whether AB CD. a.) b.) 8. Determine whether the following can be proven similar using AA, SAS, SSS. a.) b.) c.) d.) 9. The ratio of the areas of two similar figures is 4:9. What is the ratio of their perimeters? 30. Given: AD CB Prove: AE CE = DE BE 3Revised: 1/1/010 9:10 PM 4
35 10R Unit 8 Similarity CW 8.9 Solving Quadratic Equations HW: Worksheet #8.9 (following these notes) EXAMPLES: Solve for x. Express your answers in simplest radical form. 3Revised: 1/1/010 9:10 PM 5
36 10R Unit 8 Similarity HW 8.9 Solving Quadratic Equations DIRECTIONS: Solve for x. Express your answers in simplest radical form. 1. x + 5x 14 = 0. 4x 13x + 3 = 0 3. x + 7x + 3 = x + x = 0 5. x 10x +11 = x x 3 = 0 7. x 4x = 0 8. x 5 = x +10x = = x 6x 3Revised: 1/1/010 9:10 PM 6
37 10R Unit 8 Similarity - EXTRA FACTORING PRACTICE Factoring Trinomials ax + bx + c where a Practice: Factor Completely. 1, THE METHOD 1. n 11n y 4 + 5y a a 4. 5b 4 13b x + 7x v v 35 3Revised: 1/1/010 9:10 PM 7
38 Mixed Practice: Factor Completely. 1. m m x + 18x y 40y rx 4rx + 4r 4 5. x + 14x b x + 9x c x u + u x + x y + 0y z 4 5z 3 3Revised: 1/1/010 9:10 PM 8
39 Factor Completely. 1. 3m + 10m 8. 1x 8x x + 30x 75x 4. 6x + 19x x 10x x x x x + 7 3Revised: 1/1/010 9:10 PM 9
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