ࡱ> x|~#` >bjbj\.\. 4>D>Dr.)k111181ftp_`T3a,$ځhBn 17cS _7c7c g sss7cPh1s7cssz|% 1{d P}e Q{i}|60fc{6-o${1{_aLa6sa, b*_a_a_a  q&_a_a_af7c7c7c7c111D111111 CH 8 Right Triangles Radical Review 1. Simplifying  EMBED Equation.DSMT4  2. Addition  EMBED Equation.DSMT4  3. Multiplication  EMBED Equation.DSMT4  4. Division: Rationalize the denominator  EMBED Equation.DSMT4  8-1 Similarity in Right Triangles Take out your supplementary problem where you found the ratios in the similar triangles. If we label the sides with the words in regards to the larger triangle, lets write the ratios in words.  = = Geometric Mean: If a, b and x are positive numbers, with  EMBED Equation.DSMT4 , then x is the geometric mean between a and b. Ex: If NB = 16 and AN = 9, find CN, AB, BC and AC  Ex. If CN = 2 and NB = 4, find CA, AN, CB, AB EX. Lets simplify some more radicals:  EMBED Equation.3  ALGEBRA REVIEW HOMEWORK; Also do P. 288 #1-15 8.2 The Pythagorean Theorem Theorem 8-2 In a right triangle, the square of the ______________ is equal to the ______ of the _________ of the legs. What shortcuts did you discover after last nights homework? Remember the families that occur often? Find the missing sides  Ex.      4) Given isosceles trapezoid, find x  5) 6) Given the isosceles trapezoid find the altitude to the base.  7) The perimeter of a rhombus is 40 cm, and one diagonal is 12 cm long. How long is the other diagonal? 8)    HYPERLINK "../../../SKETCHPAD%20FOR%20GEOMETRY/CH%208/PYTHAGOREAN%20THEOREM.GSP"  PROOFS OF PYTHAGOREAN TH  HYPERLINK "http://www.unitedstreaming.com/search/assetDetail.cfm?guidAssetID=5DAAC059-E880-41E8-AC36-C3C2E8CBFD8C" Archaeology, distance formula 8.3 The Converse of the Pythagorean Theorem Theorem 8-3 If the square of one side of a triangle is equal to the _____ ___ ____ _______ of the other two sides, then the_________ is a __________ _________ The next two theorems allow you to determine what kind of triangle you have if it isnt a right triangle. Theorem 8-4 If c2 < a2 + b2, then m(C < 90 and (ABC is ___________ Theorem 8-5 If c2 > a2 + b2, then m(C > 90 and (ABC is ___________ Decide the type of triangle, given the sides 20, 21, 30 20, 21, 28 5, 6, 8 5, 12, 14 6, 7, 8 8-4 Special Right Triangles MEMORIZE THESE ASAP!! Theorem 8-6 In a 45(-45(-90( right triangle, the hypotenuse is ____ times as long as a ______ Label the following: Theorem 8-7 In a 30(-60(-90( right triangle, the hypotenuse is _______ as long as the shorter leg, and the longer leg is _______ times as long as the shorter leg Label the following:  Complete the following, given the picture. 1. If x = 6 then z = ______  2. If x =  EMBED Equation.3  then z= _____  3. If z =  EMBED Equation.3 then y = _____  4. If z = 10 then x = _____  1. If q = 8 then p = ______ and n = ______ 2. If n = 20 then q = _____ and p = ______ 3. If p =  EMBED Equation.3 then q = ______ and n = ______ 4. If p = 9 then q = ______ and n = ______ 5. The diagonal of a square has length 6. What is the perimeter of the square?? 6. The parallel sides of trapezoid ABCD are AD and BC. Given the sides AB,BC, and CD are each half as long as Side AD, find the size of angle D. 7. What are the angle sizes in a trapezoid whose sides have lengths 6, 20, 6, and 26? Mini-Review Find the geometric mean between 5 and 30.  Use the diagram to fill in:  x= ______  y = ______  z = ______ The sides of a triangle are given. Is the triangle acute, right or obtuse?? 3, 6, 7 4. 50, 120, 130 5. 8, 10, 13 A rectangle has length 6 and width 2. How long is each diagonal?? Find the perimeter of a square if each diagonal is 8 cm long. The perimeter of an equilateral triangle is 30 cm. Find the length of an altitude (now, what is an altitude???) 9. An isosceles triangle has sides 10, 10 and 12. How long is the altitude to the base?? Chapter 8 Review Right Triangles (1) Solve for x given each of the right triangles below, with sides as indicated.  (a) (b)  (2) Given the figure below,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , find the indicated lengths given sides as indicated.  (a) BC = 2 , CD = 8 Find: AB_____ AC_____AD_____ (b) AD = 20, AC = 12 Find: AB_____BC_____CD_____ (c) AC = 8 , CD = 16 Find: AB_____BC_____AD_____ (d) AB = 6 , CD = 5 Find: BC_____AC_____AD_____  (3) (4) Given:  EMBED Equation.DSMT4 , sides and angles as marked. Given: sides and angles as marked. Find: x______ , y______ , z_______ Find: x______. Is (A obtuse, right, or acute?  (5) The perimeter of a rhombus is 20. The length of the longer diagonal is 8. How long is the shorter diagonal?  (6) The altitude of an equilateral triangle is 12. Find the perimeter of the triangle.   (7) (8) Given: sides and angles as marked The corners are cut off a 6 inch square to form Find: AB________ , AD________ a regular octagon as shown. Find: x______  (9) (10) Given: sides and angles as marked Given: sides and angles as marked Find: x______ , y______ , z_______ Find: BC________   (11) (12)  Given:  EMBED Equation.DSMT4 , sides and angles as marked Given:  EMBED Equation.DSMT4 , sides as marked Find: AD________ and CD____________. Find: x________ Ch 8 Review Answers (1) (a) x = 26 (b) x =  EMBED Equation.DSMT4  (2) (a) AB =  EMBED Equation.DSMT4  , AC = 4 , AD =  EMBED Equation.DSMT4  (b) AB = 15 , BC = 9 , CD = 16 (c) AB =  EMBED Equation.DSMT4  , BC = 4 , AD =  EMBED Equation.DSMT4  (d) BC = 4 , AC =  EMBED Equation.DSMT4  , AD =  EMBED Equation.DSMT4  (3) x =  EMBED Equation.DSMT4  , y =  EMBED Equation.DSMT4  , z =  EMBED Equation.DSMT4  (4) x =  EMBED Equation.DSMT4  , (A is obtuse (5) 6 (6)  EMBED Equation.DSMT4  (7) AB = 13 , AD = EMBED Equation.DSMT4  (8)  EMBED Equation.DSMT4  (9) x = 10 , y =  EMBED Equation.DSMT4  , z =  EMBED Equation.DSMT4  (10) BC = 17 (11) AD = 8 , CD = EMBED Equation.DSMT4  (12) x = 24 ADDITIONAL REVIEW FOR RIGHT TRIANGLES I  1. Find x, y, z 2. Isosceles trapezoid with altitude of 6. Find the perimeter of the trapezoid. 3. Find x, y   4. Find x, y Answers 1. x= EMBED Equation.DSMT4 , y= EMBED Equation.DSMT4 , z= EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  ADDITIONAL REVIEW CH 8 RIGHT TRIANGLES II 1) Find the perimeter of the isosceles trapezoid.  2) Given:  EMBED Equation.DSMT4  sides as marked. Find: x, y, z  3) Given:  EMBED Equation.DSMT4 , sides as marked. Find: x_____, y_____, z_____ 4) Given:  EMBED Equation.DSMT4  Find: x_____, y_____, z_____.  5) Given:  EMBED Equation.DSMT4 , sides as marked. Find: x_____, y_____,  6) Given: trapezoid with sides as marked. Find: x_____, y_____.  7) Given: Three right triangles with three congruent angles. Find: v_____, w_____, x_____, y_____, z_____    8) Given: If the corners are cut off a 10 square to form a regular octagon as shown, find x_____.  9) A rhombus has a  EMBED Equation.DSMT4  angle and a 12 side. How long are its diagonals?  10) Find the perimeter of a square if one of its diagonals is  EMBED Equation.DSMT4 .  11) Given: figure as marked. Find: x_____, y_____.  12) Given:  EMBED Equation.DSMT4 BD = CD, sides as marked. Find: x_____, y_____, z_____.  13) Given:  EMBED Equation.DSMT4 , sides as marked. Find: x_____, y_____.  14) Given:  EMBED Equation.DSMT4 , sides as marked. Find x_____, y_____, z_____. Ch 8 Additional Review II Answers 1) P = 38 2) x = 36, y = 34 3) x = 12, y =  EMBED Equation.DSMT4 , z =  EMBED Equation.DSMT4  4) x =  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4  7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  9)  EMBED Equation.DSMT4  10)  EMBED Equation.DSMT4  11)  EMBED Equation.DSMT4  12)  EMBED Equation.DSMT4  13)  EMBED Equation.DSMT4  14)  EMBED Equation.DSMT4  SUPPLEMENTARTY PROBLEMS CH 8 1. Triangle ABC is a right triangle with right angle at C and an altitude from C to the hypotenuse,  EMBED Equation.DSMT4 . Name three similar triangles and set up 3 ratios for each triangle duo. There should be 3 combinations. Ex.  EMBED Equation.DSMT4 _ _ _  EMBED Equation.DSMT4  EMBED Equation.DSMT4 _ _ _ Give three proportions for these similar triangles  EMBED Equation.DSMT4 _ _ _  EMBED Equation.DSMT4  EMBED Equation.DSMT4 _ _ _ Give three proportions for the next group of similar triangles  EMBED Equation.DSMT4 _ _ _  EMBED Equation.DSMT4  EMBED Equation.DSMT4 _ _ _ Give three proportions for the third grouping of similar triangles 2. In baseball, the bases are placed at the corners of a square whose sides are 90 feet long. Home plate and second base are at opposite corners. How far is it from home plate to second base? You may leave your answer in radical form. 3. One of the legs of a right triangle is twice as long as the other and the perimeter of the triangle is 28. Find the length of all three sides. 4. Confused after grading many journals, Mrs. McGrath walked one mile due north, two miles due east, then three miles due north again, and then once more east for 4 miles. How far is Mrs. McGrath from her starting point? 5. Max and Mark were out in their rowboat one day, and Max spied a water lily. Knowing that Mark liked a mathematical challenge, Max announced that, with the help of the plant, it was possible to calculate the depth of the water under the boat. When pulled taut, the top of the plant was originally 10 inches above the water surface. When Mark help the top of the plant, which remained rooted to the lake bottom, Max gently rowed the boat 5 feet. This forced Marks hand to the water surface. Use this information to calculate the depth of the water.    6. A triangle with sides 5,12 and 13 must be a right triangle. Keeping the legs constant, how would the triangle change if the hypotenuse was lengthened to 15? To 17? 19? 7. The diagonals of a square have length 10. How long are the sides of the square? What is the ratio of the side to the diagonal? 8. The legs of an isosceles right triangle have the length s. What is the length of the hypotenuse with respect to s? 9. An equilateral triangle has a side of length s. Find the altitude to a side in terms of s. 10. What is the length of a side of an equilateral triangle whose altitude is 16?     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