Similarity #5 - Questions and Answers

Question 1 Given △ABC ~ △QRS what is the value of x? [Image of two triangles labeled ABC and QRS with side lengths labeled and an angle marked.] Answer Given that △ABC ~ △QRS by the corresponding part of similar triangles ⇒ ∠A = ∠Q, ∠B = ∠R and ∠C = ∠S now in △ADC and △QPS here ∠ADC = ∠QPS (each 90°) ∠C = ∠S (from above) using angle angle similarity criteria of triangle △ADC ~ △QPS (by angle angle similarity) by the corresponding part of similar triangles SQ/AD = QP/CA x/18 = 10/12.5 ⇒ x = (10 * 18) / 12.5 ⇒ x = 14.4 this is the required answer. Question 2 Describe one similarity and one difference between the definitions of sin θ, and cos θ, where θ is an acute angle of a right triangle. Answer Step 1 Similarity: If hypotenuse is given then using both sin θ and cos θ we can evaluate length of sides of a right triangle. Step 2 Difference: sin θ: Find the side opposite to the acute angle θ cos θ:Find the adjacent side to the acute angle θ Question 3 Compare and contrast the Borda count method and the plurality-with-elimination method. Identify at least one similarity and one difference. Use a specific example of voting preferences to illustrate your point. Answer Step 1 Borda count and plurality elimination method are the two voting methods used widely. Borda count is an example of single winner election method.in borda count method the voters arrange the candidates in order of preference. In this method the winner is determined by giving each candidate, some points which depends upon the number of candidates ranked lower. And after all the votes are counted the option or candidate with most points is announced as the winner of the election. Step 2 On the other hand, in plurality and elimination method, the candidates who have majority of first place votes is the winner. The voting is carried out in multiple rounds if there is no majority for any candidate. In this method the first round is to count the first-place votes and then eliminating the candidates who secured lowest votes from the competition. And their names are crossed out from the list of candidates. Then again, the first-place votes are counted and the candidate with majority of first place vote is announced as winner if there is such a candidate. otherwise the candidate with fewest first place vote is eliminated. Then the process is repeated until there is a winner with majority of first place votes. Step 3 The difference between the two is in borda count the voters can rank or vote for all the possible candidates but in the plurality and elimination method one voter can only vote for one candidate. In both the borda count and plurality and elimination methods the candidate with most points or vote wins. Question 4 Determine whether the triangles are similar. If so, write a similarity statement and name the postulate or theorem you used. If not, explain. [Image of two triangles labeled LNO and LMN] Answer Step 1 given we have been given a triangle MNL in which P is a point on the side MN and O is the point on the side NL and OP is parallel ML Step 2 first, we will take △NPO and △NML ∠N = ∠N (common angle) ∠NPO = ∠NML (corresponding angles as OP||ML) so, the AA rule which states in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar. △NML ≈ △NPO (AA rule) therefore the two triangles NML and NPO are similar. Question 5 Triangle MPT with NR||MT is shown below. The dimensions are in centimeters. [Image of a triangle labeled MPT with a line segment NR parallel to MT] Which measurement is closest to the length of RT in centimeters? 1)1.4 2)1.8 3)3.4 4)4.3 Answer Step 1 Given, △MPT with NR||MT PN = 4 cm PR = 2.5 cm NM = 2.9 cm We have to find the length of RT in centimeters. Step 2 We have NR||MT Then, using the properties for two parallel lines cut by transversal lines ∠PNR = ∠PMT (1) ∠PRN = ∠PTM (2) Now, In △NPR and △MPT ∠PNR = ∠PMT (from (1)) ∠PRN = ∠PTM (from (2)) ∠NPR = ∠MPT (common angle) So, From AAA (angle-angle-angle) rule of similarity △NPR~△MPT Step 3 Now, In similar triangle, ratio of corresponding sides will be equal. NP/MP = PR/PT = NR/MT Taking NP/MP = PR/PT 4/(2.9 + 4) = 2.5/(2.5 + RT) (substituted the values) ⇒ 4(2.5 + RT) = 6.9 × 2.5 ⇒ 4RT + 10 = 17.25 ⇒ 4RT = 17.25 - 10 ⇒ 4RT = 7.25 ⇒ RT = 7.25/4 ⇒ RT = 1.8125 So, The lenght of RT = 1.8cm