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-1- Name:_____________________________ Geometry Rules! Period:_______ Chapter 7 Notes Notes #36: Solving Ratios and Proportions and Similar Triangles (Sections 7.1 and 7.2) 3 , 3 to 4, 3:4 4 3 6 Proportion: two ratios that are equal to each other. 4 8 Ratio: a comparison of two quantities. A. Solving Proportions Cross-multiply and set the products equal to each other Be sure to use FOIL when you are multiplying binomials together Solve for the variable Solve for x: 1.) 2 3 x 12 2.) 3.) x3 4 2 3 4.) x 7 3 5 x 1 x 5 2x 3 2x -2- B. Properties of Proportions Complete: 5.) If x 3 , then 7 x ____ 2 7 7.) If a : 2 5 : 3, then 3a ____ 9.) If x y x , then ____ 5 2 y 11.) For the given figure, it is given that: KR = 6, KT = 10, KS = 8 6.) If 5 3 , then 3x ____ x 2 8.) If x 2 y , then ____ y 9 x 10.) If x y x3 , then ____ 3 4 3 KR KS . Solve for the missing lengths. RT SU RT = ____ K SU = ____ R S T KU = ____ U C. Similar Polygons Similar polygons have the same _________ but not necessarily the same ________. Example of similar triangles: B Y 6 A 3 10 X 8 5 4 Z C Their corresponding angles are ___________________ Their corresponding sides are in a ______________ _______________________ This ratio is called a ____________ ______________ and in this case is _______ We show that they are similar with this statement: ___________________ -3- 12.) ABCDE A' B ' C ' D ' E ' a) scale factor = _______ 9 C x 160 D b) mA ' _____ , mD _____ mC ' _____ B 6 100 A C' y 30 2 c) x = _____, y = ______, z = ______ E 8k D' B' 4 A' E' z The figures are similar. Solve for the variables. (Hint: redraw the diagram as two figures) 13.) 18 12 10 y x z 16 8 14.) 12 16 x y 9 24 -4- D. Algebra Practice: Factor: 15.) 4 p 4 r 3s 2 16 p 2 r 3s 4 16.) 3x 2 15 x 21 17.) 2 x3 8 x 2 8 x Simplify: 18.) 3 15 19.) 2 8 3 12 20.) 3 5 22.) 2 x 2 10 x 28 23.) 12 x 2 7 x 10 2 Solve: 21.) 3x 2 5 x 2 0 -5- Notes #37: Similar Triangles (Sections 7.3 and 7.5) Similar triangles have: ________________ corresponding angles sides that are in __________________ _______________________ You can conclude that two triangles are similar if: _________: two pairs of corresponding angles are congruent _________: all three pairs of sides are in the same proportion _________: two pairs of sides are the same proportion and their included angles are congruent -6- Are the triangles similar? If so, state the similarity and the postulate you used. Re-draw the triangles in matching positions Mark congruent angles Test sides for a constant proportion: small medium l arg e small medium l arg e Look for these patterns: AA~, SSS~, SAS~ 1.) 2.) F B 35 60 O D 85 E N P 60 A C Q M 3.) 4.) B F E 15 4 4 6 6 8 B F D C 10 D 12 A C A 5.) 6.) R 5 4 8 10 Q S 6 70 9 X 10 70 Y 7.5 6 Z 15 -7- State whether the figures are always, sometimes, or never similar: do they always, sometimes, or never have the exact same shape? 7.) two squares 8.) two congruent triangles 9.) two rectangles 10.) two rhombuses 11.) two pentagons 12.) two regular octagons Proportional Lengths (Section 7.5) A. Triangle Proportionality A parallel slice cuts a triangle’s sides proportionally ( Side-Splitter Theorem) a j c k , a c a c , a j d b Example: Solve for x 12 33 a b 20 x d k , j b -8- B. Angle Bisector Proportionality An angle bisector proportionally divides the opposite side z w y z y w z x x w Example: Solve for x: x 21 8 14 C. Parallel Line Proportionality Parallel lines proportionally divide their transversals a a b b d c d c Example: Solve for x: 18 9 24 - x x a c -9- Solve for the variables: 1.) 2.) 14 9 6 24 18 4 x x 60 3.) 4.) 20 x x 12 18 21 x+ 2 5.) 6.) 3 16 12 12 2x 7.5 4 25 - 10 - 13.) Write the equation of a line that contains the point ( -4, 3) and has a slope of . 15.) Write the equation of a line in standard form that is parallel to 3 x 6 y 5 and contains the point ( 1, 4). 16.) Write the equation of a line that contains ( -2, -3) and ( 4, -9) in standard form. 18.) Write the equation of a line that is perpendicular to y 2 x 5 and contains the point ( -6, 7) 1 2 - 11 - Notes #38: Similarity in Right triangles ( 7.4) Geometric Means and Similar Right Triangles A. Geometric Mean asks the question: “what number, squared, equals the product of two given numbers?” Find the geometric mean of the listed numbers: Use the given numbers in this equation: x2 = ab Solve for x 1.) 9 and 16 2.) 12 and 3 3.) 5 and 15 B. Similar Right Triangles When an altitude of a right triangle is drawn to its hypotenuse, three similar right triangles are formed: y x a y a(a b) z b(a b) x (a)(b) z b - 12 - Solve for the variables: Re-draw the three triangles and label all sides Set up proportions to solve for the variables Look for ways to use the Pythagorean theorem 4.) p n m 5 20 m (5)(25) p (20)(25) n (5)(20) 5.) 1 4 a 1 9 b c - 13 - 6.) 3 5 y x z - 14 - Notes #39—Section 8.1 Pythagorean Theorem In Words: Pictures/Symbols: Example: Find the missing side of the triangles below. 1.) In a _____________ triangle, the sum of the __________ of the lengths of the _________ is equal to the __________ of the length of the ______________. x 5 12 2.) 5 x 3 A Pythagorean Theorem. is a set of whole numbers a, b, and c, that satisfy the Examples: Do the lengths of the sides given form a Pythagorean triple? 3.) 8, 15, 17 4.) 7, 4, 6 5.) 20, 21, 29 Examples: Find the value of x. Leave your answer in simplest radical form. 6.) 7.) 16 12 x 34 10 8.) 16 x 12 - 15 - Determining Whether a Triangle is Right, Acute, or Obtuse Given Three Side Lengths: Right Acute Obtuse Ex 9: Sides have lengths 3, 4, and 5 Ex 10: Sides have lengths 12, 6, and 11 Ex 11: Sides have lengths 14, 7 and 12 Examples: The lengths of the sides of a triangle are given. Classify the triangle as acute, right, or obtuse. 12.) 10, 15, 20 13.) 7, 6, 4 14.) 15, 20, 25 Examples: Find the value of x. Leave your answer in simplest radical form. 15.) 16.) 22 22 x 2 3 x 36 8 2 - 16 - Algebra Review: Solve using quadratic formula Examples: 17.) x 2 4 x 5 0 18.) 2 x 2 5 10 x 19.) 3x 2 8 x 4 20.) 4 x 2 6 x 2 - 17 - Notes #40: Chapter 7 Review Simplify each ratio: 1.) a) BC:CD b) m<B:m<C B 8 C 6 c) CD: Perimeter of ABCD A 40 2.) If x = 4, y = 6, z = 2 find each ratio: a) x to y 6a 2 b 5 3.) 12ab7 b) (x + z) to y c) 4.) x y 7z 2x y for x 3, y 2, z 1 zx Write and equation and solve: 4.) The ratio of the angles of a triangle is 1:3:5. Find the angles. 5.) The ratio of the angles of a pentagon is 6: 8: 9: 11: 11. Find the angles. D - 18 - Are the triangles similar? If so, write a similarity statement and the postulate you used: 6.) 7.) C 6 B 10 16 8 15 A 10 E 12 D 4 8.) 9.) E N 12 9 D P O F 15 Q Y M 30 18 X 24 Z Solve for the variables: 10.) x 1 x 4 x 3 x 8 11.) x 12.) y 3 12 7 5 15 22.5 x 20 - 19 - 13.) 14.) 20 16 12 8 x 24 12 - x Simplify: 15.) Similar Right Triangles: Solve for m, n, and p in reduced radical form. p n m 5 x 16.) a.) Find the geometric mean of 5 and 10 b.) Find the geometric mean of 4 and 20. 10 Are the figures sometimes, always, or never similar? 17.) two rectangles 18.) two equilateral triangles 19.) two regular hexagons - 20 STUDY GUIDE 7 Name:________________________ Show all your work! Date:____________Period:_______ For #1-3, ABCD is a parallelogram. Simplify each ratio: 1.) BC:CD 1.) ___________ 12 A B 2.) AD:(Perimeter of ABCD) 2.) ___________ 8 120 3.) mA : mB C D 3.) ___________ For #4 – 6, complete each statement: 4.) If a : 3 = 7 : 4, then 4a = _____ 5.) If x 3 , then y 8 6.) If x y , then 2 3 x2 2 y x 4.) __________ 5.) __________ 6.) __________ For #7-10, solve for x: 7.) x 15 10 25 8.) x2 4 x3 5 7.) x = ______ 8.) x = ______ 9.) x 1 x4 x2 x2 10.) 3 4x 1 1 5 x 2 3x 9.) x = ______ 10.) x = ______ For #11 – 16, state whether the two polygons are always, sometimes, or never similar. 11.) two right triangles 12.) two scalene triangles 13.) two squares 14.) two rectangles 11.) __________ 12.) __________ 13.) __________ 14.) __________ 15.) an isosceles triangle and a right triangle 16.) two regular hexagons 15.) __________ 16.) __________ - 21 For #17 - 20, refer to the diagram. 17.) Find mM ' 17.) __________ A 6 18.) __________ 18.) Find the scale factor of MATH to M’A’T’H’ T x 70 M 18 4 19.) Solve for x. H 19.) x = ______ A' T' 2 H' y M' 20.) y = ______ MATH ~ M’A’T’H’ 20.) Solve for y. For #21 – 34, complete the similarity statement and state why the triangles are similar. If the triangles are not similar, circle not similar. (If you are using SAS similarity or SSS similarity, be sure to check your side lengths for a common proportion) 21.) 21.) QRS _____ by __________ 22.) OR N R P 9 6 Q not similar O S 12 16 X 12 Y Q M 22.) MNO _____ by __________ 8 Z OR not similar - 22 23.) 24.) 23.) ABC _____ by __________ F A 12 B 21 20 8 OR C C B D 6 not similar 30 31.5 9 D A E 24.) ABC _____ by __________ OR not similar For #25 – 28, solve for x and y (where x and y are positive): 25.) 26.) 12 25.) x = ______ y = ______ 16 6 7 x y 9 x 12 26.) x = ______ 24 27.) 28.) 27.) x = ______ 15 12 x 5 24 10 9 28.) x = ______ x Solve: 29.) 6 x 2 16 x 6 30.) 4 x 2 12 x 40 0 31.) 3 x 3 48 x 32.) 3x 2 5 9 x 29.) _______ 30.) _______ 31.) _______ 32.) _______ - 23 For #33-34, find the geometric mean of the two numbers. 33.) 5 and 10 34.) 4 and 20 33.) ________ 34.) ________ For #35-36, solve for x, y, and z. (Hint: use 3 similar, right triangles) 35.) x 35.) x = ______ y = ______ z y z = ______ 4 25 36.) 36.) x = ______ y y = ______ z 4 z = ______ x 16 For # 37-38, solve for x. Leave the answer in simplified radical form 37) 38.) 37.) x = ______ x 12 x 38.) 6 x = ______ 9 For #39-40, the lengths of the sides of a triangle are given. Classify each triangle as acute, right, or obtuse. 39.) 20, 30, 40 40.) 41, 9, 40 39.)____________ 40.)__________