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Date: _________________________ AIMS Practice Problems 6 1. If the pattern below continues, what will be the 8th term? 4. If RS = 50.1 and QS = 82.9, find QR. 2.2, 2.5, 2.8, 3.1, 3.4, … A. B. C. D. 4.0 4.1 4.2 4.3 2. Determine the relationship between the two equations yx 4 y 4 4x Q A. B. C. D. R S 32.8 133 50.1 22.8 5. What is the slope of the line in the figure below? 5 4 A. Parallel B. Perpendicular C. Coincide D. Intersecting but not perpendicular 3 2 1 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 3. The set {-3, 0, 6, 12} is a subset of A. Whole Numbers B. Integers C. Natural Numbers D. Irrational Numbers A. 4 B. -4 C. 1 4 D. 1 4 2 3 4 5 NOTES: 7.1 Ratios and Proportions Proportion - an equation that states that two ratios are equal *To solve: cross multiply and solve. Solve the proportion. 1. 3 6 x 8 2. 5 15 3 y 3. x 5 18 6 4. 4 x 9 63 5. 5 y2 3 6 6. m 2 14 5 10 7. 2 8 7 w2 8. 12 96 7 y2 Ratio and Proportion Ratio: a comparison of two numbers *Ratio a to be can be written in three ways a b a :b a to b Simplifying ratios: make sure units are the same (if not convert) simplify the fraction to lowest terms Simplify the ratio (like you would simply FRACTIONS) 1. 60cm : 200cm 2. 3 ft 18 ft 3. 12m 10m 4. 2months 8months Use the diagram to answer the following questions. A B C 1 2 3 4 5 6 D E 7 8 9 10 11 12 13 14 Use the number line to find the given ratio. 1. AB : BC 2. BD : DE 3. AC : AE 4. BD : BE 5. AD : AE 6. EC : CA Using ratios 1. In the diagram below AB : BC is 4 :1 and AC 30 Find AB and BC A B C 2. FG : GH is 4 : 3 , FH 56 Find FG and GH 3. The perimeter of a rectangle is 80 ft . The ratio of the length to width is 7 : 3 Find the length and the width. 4. In the diagram EF : FG is 2 :1 and EG 24 Find EF and FG E F G 5. The perimeter of a rectangle is168 ft . The ratio of the length to width is 4 : 3 . Find the length and the width of the rectangle Assignment: Section 7.1 Page 361- 363 # 25-45 NOT 27, and 55 Date: _________________________ AIMS Practice Problems 6 1. The square root of 31 belongs to which subset? A. B. C. D. Natural Numbers Counting Numbers Rational Numbers Irrational Numbers 2. Which value is closest to A. B. C. D. ? A. B. C. D. x=5 x = 18 x = -18 x = -5 8.1 9.2 10.8 12.1 3. Evaluate 3x - 4y for the values x = -4 and y = -2. A. B. C. D. 4. Which of the following values of x makes the proportion below true? -20 -4 4 20 5. Which of the following is an example of dependent events? A. flipping a fair coin twice and getting tails both times B. choosing the starting player line-up for a basketball game C. choosing two cards from a stack of colored cards, with replacement, and both cards are blue D. rolling a 6-sided die 2 times and getting a 5 both times NOTES: 7.2 Similar Polygons Polygons are similar when 1. corresponding (same place) angles are congruent (the same) 2. corresponding (same place) sides are proportional (reduce to same fraction) is the symbol for similar ABCD EFGH . List corresponding angles and ratios of corresponding sides. C G B F A D E H Statement of Proportionality Corresponding Angles (Ratio of Corresponding Sides) Scale Factor: the SIMPLIFIED ratio (fraction) of the lengths of two corresponding sides of a similar polygon ** if all fractions reduce to the same thing, they are not only similar, but this is also your scale factor. In the diagram PRQ STU T R 16 8 20 S 10 P 15 12 U Q 1. List all pairs of congruent angles. 2. Write the statement of proportionality. 3. Check that the ratios of the sides are equal (use the ratios of sides from above, only use the numbers, not the letters) 4. What is the scale factor of the above figures? Determine whether the polygons are similar. are corresponding angles ? are corresponding sides proportional? If similar, write a statement of similarity ( ABC XYZ ) a AB BC CA statement of proportionality ( ) and find the XY YZ ZX scale factor. H K 1. 9 12 16 G M 2. 12 L 20 12 N J 15 P 10 S 9 T 6 Q U V R E 3. Y 12 18 6 9 Z 12 X F D 8 Solving Similar Polygons Solve for variables of similar polygons by using corresponding sides to set up a proportion. Solve for the variables. 1. RST GHJ S H 15 10 9 8 T R x 2. PQR J G y STU T Q 7 14 x S P 9 16 y R U 3. ABCD WXYZ W A X B b 4 D 4. C 6 XYZ Z Y 12 DEF E Y 27 x 21 28 Z F D 20 X Assignment: Section 7.2 Page 368 – 371 # 8 – 26 even y Date: _________________________ AIMS Practice Problems 6 1. Simplify: ( x 4)( x 3) A. B. C. D. 2. If RS = 41.7 and QS = 77.8, find QR. x 2 x 12 x 2 7 x 12 x2 1 Q A. B. C. D. x 1 R S 36.1 26.1 119.5 41.7 1 3. Which of the graphs below correctly represents the equation y x 3 ? 2 A. B. -5 -4 -3 -2 5 5 4 4 3 3 2 2 1 1 -1 C. 1 2 3 4 5 -4 -3 -2 -4 -3 -2 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 D. 5 -5 -5 -1 4 3 3 2 2 1 1 1 2 3 4 5 2 3 4 5 1 2 3 4 5 5 4 -1 1 -5 -4 -3 -2 -1 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 Review: Are the polygons similar? If so, write the similarity statement, the statement of proportionality and find the scale factor. Y B 12 6 2 A 4 8 C X 16 Solve the proportion. 1. 5 y2 3 6 2. 3 6 x 8 3. 5 15 3 y 4. m 2 14 5 10 Z NOTES: 7.3 Showing Triangles are Similar Angle – Angle (AA) Angle – Angle Similarity Postulate (AA) If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar Y K J L X Z JKL XYZ Determine whether the triangles are similar. If they are, write the similarity statement. you may need to find the missing angle measurement. Subtract from 180. K 1. H 61 G J 29 L R 2. 80 T 65 S M F 80 L 39 N Are you given enough information to determine whether the triangles are similar. If so, write the similarity statement. shared angles are vertical angles are 1. R M 80 80 T 65 S R 2. T S 48 U 35 L V 48 3. A D B C E N Using Similar Triangles Write a similarity statement for the triangles. Then find the value of x. 1. H K 25 x G F 40 2. 28 L T Q 3 5 x S P 3. C A x B 6 y R 15 12 2 D 6 E U 4. G 6 12 H 4 x I 5. S K J 16 T 45 8 R x U 45 4 V Assignment: Section 7.3 Page 375 - 376 # 2 – 26 even Date: _________________________ AIMS Practice Problems #53 6 1. Draw a labeled diagram for a line segment. A. 3. The n th term of the pattern defined in the table is given by which expression? 2 4 C B 3 4 5 n 9 16 25 ? A. 5n B. n 2 C. n 2 D. n 5 A B. A B C. A B 4. If mCOD = 24 and mBOC = 20 , then what is the measure of BOD ? D. B A B C D O 2. Which of these equations represents a line passing through (5,3) and ( 5, 3) ? A. B. C. D. 3 x 5 5 y x 3 y 5 y 3 y A. B. C. D. 41° 42° 49° 44° NOTES: 7.4 Showing Triangles are Similar Side – Side – Side (SSS) Side – Angle – Side (SAS) Side – Side – Side Similarity Theorem (SSS) If the corresponding sides of two triangles are proportional, then the triangles are similar Determine if the triangles are similar. If they are, write the similarity statement and find the scale factor. start with the shortest sides R 1. 8 12 T S 10 M 6 4 L 2. A N 5 12 6 C 9 B 14 G 6 J 10 H Side – Angle – Side Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides that include these angles are proportional, then the triangles are similar. Determine if the triangles are similar. If so, write the similarity statement. D 1. A 5 3 60 B 2. 3 C 60 5 E F L G 6 8 M H J 12 3. R D 6 85 85 8 E N 8 10 S 8 T F Similarity in Overlapping Triangles Show that the overlapping triangles are similar Separate the triangles and label or mark EVERYTHING 1. V 5 4 X W 10 8 Y Z N 2. 40 M 16 P 35 Q 14 L 3. P 5 3 R Q 5 3 S T Assignment: Section 7.4 Page 382 – 385 # 5 – 9, 15 – 25 ONLY ODD # 32, 33, 38 – 41 Date: _________________________ AIMS Practice Problems 6 1. Determine the solution to the systems of equations represented by the following graph. 5 3. If mCOD = 27 and mDOE = 26 , then what is the measure of COE ? 4 3 2 C 1 -5 -4 -3 -2 -1 1 2 3 4 D 5 -1 E -2 O -3 -4 A. B. C. D. -5 A. B. C. D. (0,1) (1, 0) (0, 0) (1, 1) 53° 58° 51° 50° 4. Which is the formula to find the slope of a line? 2. If RS = 30.4 and QS = 60.4, find QR. A. B. Q A. B. C. D. R 30 90.8 20 30.4 S C. D. x2 x1 y2 y1 y y m 2 1 x2 x1 y y1 m 2 x2 x1 x x m 2 1 y2 y1 m Practice: Determine whether the triangles are similar. If they are similar, state the similarity postulate or theorem that justifies your answer. R 1. 2. 80 R D 6 85 85 8 T 65 10 8 S T S M E F 80 39 L N R 3. 4. R 8 12 T T S 48 6 4 U L V 48 S 10 M N 5 N K 5. 6. H 40 61 J 29 M G 16 P 35 Q 14 L F L Find the value of the variable. 7. 8. NOTES: 7.5 Proportions and Similar Triangles Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides then it divides the two sides proportionally R RT RU TU QS so TQ US T U Q S Find the value of the variable 1. set up the proportion, go from small to big 2. plug in side length values 3. solve for the variable 1. R 4 x T U 12 8 Q 2. S R 9 Q 3 S y 20 P 3. 4 x 10 5 4. 14 y 15 6 Determine if the segments are parallel. are the RATIOS of the sides EQUAL? 1. MN GH ? 21 G M 56 L 2. 48 N 16 H TS RQ ? T 21 R 15 S 3. Q 23 P 17 QR ST ? P 6 4 R Q 12 8 S T The Midsegment Theorem The Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long In ABC , if CD DA and CE EB then 1 DE AB and DE AB 2 C D E A B Find the value of the variable 1. w 2. 17 x 10 q 3. 4 4. 3 8 p 4 11 14 3 8 11 16 Assignment: Section 7.5 Page 390 – 392 # 3 – 29 odd, 44 – 52 even 6 Date: _________________________ AIMS Practice Problems 1.The endpoints of a line segment are the points with the coordinates (2,1) and (8,9). What are the coordinates of the midpoint of this line segment? A. (2,3.5) B. (3,4) C. (5,5) D. (4.5,5.5) 2. Which property is demonstrated below? If A=2c and 2c=D, then A=D. A. Associative Property B. Distributive Property C. Reflexive Property D. Transitive Property 3. The numbers in the data set below can be classified as members of which subset of real numbers? A. B. C. D. Integers Irrational Numbers Rational Numbers Whole numbers 4. Paul's first project scores for art were: 96, 87, 84, 96, 100 Which statement is true about the project scores? A. The mode is the same as the median. B. The median is the same as the mean. C. The range is the same as the mode. D. The mode is the same as the mean. Review: Determine if the following triangles are similar. If they are, write the similarity statement. Y 1. K 33 L J 32 Z R 2. T S 60 U V 60 D 3. A 3 4 B 5 6 C 3 4. A F 5 E 8 L J 110 6 8 K 110 C 10 B X 5. Solve for the variable Q 10 y U R 36 14 S T 6. Determine if CD is parallel to EF B 72 96 D C 40 30 E F Find the value of the variable 7. 8. x w 13 40 Assignment - REVIEW: Page 401 – 403 # 1 – 23 ALL