Similarity #1 - Questions and Answers

Question 1 Determine the length of x in the diagram below. [Image of a diagram with two triangles labeled ABC and ADE with side lengths labeled and angles labeled as well.] Answer Step 1 We have to calculate the value of x. Step 2 In the given figure we have two triangles △ABC and △ADE [Image of a diagram with two triangles labeled ABC and ADE with side lengths labeled and angles labeled as well.] ∠BAC = ∠DAE = common angle ∠ADE = ∠ABC = congruent angles (As BC || DE) using the rule of angle - angle similarity two triangles △ABC and △ADE are similar. Now taking the ratio AD/AB = DE/BC 25/60 = 12/x x = 12 * 60 / 25 x = 28.8 Question 2 Identify a similarity statement for the two triangles. The triangles are not similar △TRS ~ △LJK △RST ~ △JKL △SRT ~ △KLJ △TSR ~ △LJK S 8 112° R L 24 18 T 112° 10 K [Image of two triangles labeled TRS and LJK with side lengths and an angle labeled] Answer Step 1 we will check whether two traingle satisfy SAS criterion of similiarity Step 2 Now RT/JL = 18/24 = 3/4 and ST/KL = 8/10 = 4/5 RT/JL ≠ ST/KL ⇒ It does not satisfy SAS criterion ⇒ These two triangle are not similiar. Question 3 Can the triangles below be proven similar by the SAS Similarity Postulate? [Image of two triangles labeled ABC and DEF with side lengths and an angle labeled] Yes, the SAS Similarity Postulate can be used because 2 sets of corresponding sides are proportional, and the included angles by these sides are congruent. Yes, the SAS Similarity Postulate can be used because 2 sets of corresponding sides are congruent, and the included angles by these sides are congruent. No, the SAS Similarity Postulate cannot be used because the corresponding angles are congruent but not included by the proportional sides. No, the SAS Similarity Postulate cannot be used because the 2 sets of corresponding sides are not proportional. Answer Step 1 here we have triangles △ABC and △EDF and if we use the SAS similarity postulate we have to make sure to use any two sides and one angle from each triangle respectively such as AB/ED = BC/DF and ∠B = ∠D or AB/DF = BC/ED and ∠B = ∠D Step 2 using either of any two ratios we get 6/5 = 9/2.5 and ∠B = ∠D = 67° but 1.2 ≠ 4.5 or 6/2 = 9/5 and ∠B = ∠D = 67° but 3 ≠ 1.8 Step 3 so, we can see that neither of them is satisfying the SAS rule as their corresponding angles are equal but their corresponding sides are not proportional. therefore, the correct option is D Question 4 If A, B, and C are the vertices of a triangle, find AB+BC+CA Answer [Image of graph of triangle vertices at A(-1,1), B(1,3), C(3,1) with area shaded in blue.] The graph is above Therefore, Area is 4. The total Area can be divided into two parts. Part 1: Area of the red triangle = Base * Height/2 = 2 * 2 / 2 = 2 Part 2: Area of two blue squares of side length 1 unit. Area of one square is 1 square unit. Therefore, area of two squares is 2 square units. Therefore, ∫f(x) dx = Area of the Region below f(x) and above x axis = 2+2 = 4 Result ∫(from 0 to 4) f(x) dx = 4 Question 5 Can the triangles below be proven similar by the SSS Similarity Postulate? Explain. [Image of two triangles labeled ABC and PQR with side lengths and an angle labeled] No, the SSS Similarity Postulate cannot be used to prove the two triangles are similar. To use the SSS Similarity Postulate, all three corresponding sides must be known to be proportional. 2)Yes, the SSS similarity Postulate can be used to prove the two triangles are similar. The triangles have one set of congruent corresponding angles, so the SSS Similarity Postulate can be used. 3)No, the SSS Similarity Postulate cannot be used to prove the two triangles are similar. To use the SSS similarity Postulate, all three corresponding angles must be known to be congruent. No, the SSS Similarity Postulate cannot be used to prove the two triangles are similar. To use the SSS Similarity Postulate, all three corresponding sides must be known to be congruent. Answer 1 No, the SSS Similarity Postulate cannot be used to prove the two triangles are similar. To use the SSS Similarity Postulate, all three corresponding sides must be known to be proportional. Answer 2 Step 1 We have the given figure as [Image of two triangles labeled ABC and PQR with side lengths and an angle labeled.] Here, all three sides of both the triangles are missing. So we can not predict the corresponding ratio of sides of both of the triangles. Step 2 SSS stands for "side-side-side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio. Hence, option (1) is correct option.