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1 Using Similar Right TrianglesSay Thanks to the AuthorsClick (No sign in required)To access a customizable version of this book, as well as otherinteractive content, visit Foundation is a non-profit organization with a mission toreduce the cost of textbook materials for the K-12 market both inthe and worldwide. Using an open-source, collaborative, andweb-based compilation model, CK-12 pioneers and promotes thecreation and distribution of high-quality, adaptive online textbooksthat can be mixed, modified and printed ( , the FlexBook textbooks).Copyright 2015 CK-12 Foundation, names CK-12 and CK12 and associated logos and theterms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and form of reproduction of this book in any format or medium,in whole or in sections must include the referral attribution (placed in a visible location) inaddition to the following as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordancewith the Creative Commons Attribution-Non-Commercial (CC BY-NC ) License ( )

2 , as amended and updated by Creative Com-mons from time to time (the CC License ), which is incorporatedherein by this terms can be found at : April 3, 1. Using Similar Right TrianglesCHAPTER1 Using Similar RightTrianglesLearning Objectives Identify similar triangles inscribed in a larger triangle. Evaluate the geometric mean. Find the length of an altitude or leg using the geometric Queue1. If two triangles are right triangles, does that mean they are similar? If two triangles are isosceles right triangles, does that mean they are similar? Solve the ratio:3x= If the legs of an isosceles right triangle are 4, find the length of the hypotenuse.

3 Draw a picture and simplifythe What?In California, the average home price increased in 2004 and another in 2005. What isthe average rate of increase for these two years?Inscribed Similar TrianglesYou may recall that if two objects are similar, corresponding angles are congruent and their sides are proportional inlength. Let s look at a right triangle, with an altitude drawn from the right are three right triangles in this picture,4 ADB,4 CDA, and4 CAB. Both of the two smaller triangles aresimilar to the larger triangle because they each share an angle with4 ADB. That means all three triangles are similarto each 8-5:If an altitude is drawn from the right angle of any right triangle, then the two triangles formed aresimilar to the original triangle and all three triangles are similar to each proof of Theorem 8-5 is in the review 1:Write the similarity statement for the triangles :Ifm6E=30 , thenm6I=60 andm6 TRE=60.

4 M6 IRT=30 because it is complementary up the congruent angles in the similarity 4 ITR 4 RTEWe can also use the side proportions to find the length of the 2:Find the value :First, let s separate the triangles to find the corresponding we can set up a leg in4 EDGshorter leg in4 DFG=hypotenuse in4 EDGhypotenuse in4 DFG6x=10848= 3:Find the value 1. Using Similar Right TrianglesSolution:Let s set up a leg in4 SVTshorter leg in4 RST=hypotenuse in4 SVThypotenuse in4 RST4x=x20x2=80x= 80=4 5 Example 4:Find the value :Use the Pythagorean +(4 5)2=202y2+80=400y2=320y= 320=8 5 The Geometric MeanYou are probably familiar with the arithmetic mean, whichdivides the sumofnnumbers byn.

5 This is commonlyused to determine the average test score for a group of geometric mean is a different sort of average, which takes thenthroot of the productofnnumbers. In this text,we will primarily compare two numbers, so we would be taking the square root of the product of two numbers. Thismean is commonly used with rates of increase or Mean:The geometric mean of two positive numbersaandbis the numberx, such thatax=xborx2=abandx= 5:Find the geometric mean of 24 and :x= 24 36= 12 2 12 3=12 6 Example 6:Find the geometric mean of 18 and :x= 18 54= 18 18 3=18 3 Notice that in both of these examples, we did not actually multiply the two numbers together, but kept them made it easier to simplify the practical application of the geometric mean is to find the altitude of a right 7:Find the value.

6 Using similar triangles, we have the proportionshortest leg of smallest4shortest leg of middle4=longer leg of smallest4longer leg of middle49x=x27x2=243x= 243=9 3In Example 7,9x=x27is in the definition of the geometric mean. So, the altitude is the geometric mean of the twosegments that it divides the hypotenuse 8-6:In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuseinto two segments. The length of the altitude is the geometric mean of these two 8-7:In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuseinto two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuseand the segment of the hypotenuse that is adjacent to the 8-6:BCAC=ACDCT heorem 8-7:BCAB=ABDBandDCAD=ADDBBoth of these theorems are proved using similar 8:Find the value 1.

7 Using Similar Right TrianglesSolution:Use theorem 8-7 to solve 35y2=15 35x= 20 35y= 15 35x=10 7y=5 21 You could also use the Pythagorean Theorem to solve fory, oncexhas been solved for.(10 7)2+y2=352700+y2=1225y= 525=5 21 Either method is What? RevisitedThe average rate of increase can be found by using the geometric the two year period, housing prices increased QuestionsUse the diagram to answer questions Write the similarity statement for the three triangles in the IfJM=12 andML=9, the geometric mean between the following two numbers. Simplify all 16 and 326. 45 and 357. 10 and 148. 28 and 429. 40 and 10010. 51 and 8 Find the length of the missing variable(s).

8 Simplify all 1. Using Similar Right Write a proof for Theorem :4 ABDwithAC DBand6 DABis a right :4 ABD 4 CBA 4 CAD21. Fill in the blanks for the proof of Theorem :4 ABDwithAC DBand6 DABis a right :BCAB= DBand6 DABis a right 4 CBA Last year Poorva s rent increased by 5% and this year her landlord wanted to raise her rent by What isthe average rate at which her landlord has raised her rent over the course of these two years? Mrs. Smith teaches AP Calculus. Between the first and second years she taught the course her students average score improved by 12%. Between the second and third years, the scores increased by 9%. What is theaverage rate of improvement in her students scores?

9 24. According to the US Census Bureau, the rate of growth ofthe US population was and in 2009 it was What was the average rate of population growth duringthat time period?Algebra ConnectionA geometric sequence is a sequence of numbers in which each successive term is determinedby multiplying the previous term by the common ratio. An example is the sequence 1, 3, 9, 27, .. Here each term ismultiplied by 3 to get the next term in the sequence. Another way to look at this sequence is to compare the ratiosof the consecutive Find the ratio of the 2ndto 1stterms and the ratio of the 3rdto 2ndterms. What do you notice? Is this true forthe next set (4thto 3rdterms)?26. Given the sequence 4, 8, 16.

10 , if we equate the ratios of the consecutive terms we get:84=168. This meansthat 8 is the _____ of 4 and 16. We can generalize this to say that every term in a geometricsequence is the _____ of the previous and subsequent what you discovered in problem 26 to find the middle term in the following geometric 5, ____, 2028. 4, ____, 10029. 2, ____,1230. We can use what we have learned in this section in another proof of the Pythagorean Theorem. Use thediagram to fill in the blanks in the proof +eanddb=b?Theorem (d+e)andb2=d(d+e)? +b2=?Combine equations from # ?Distributive +e?6. ?Substitution PoEReview Queue Answers1. No, another angle besides the right angles must also be 1.