Similarity #4 - Questions and Answers

Question 1 Write the similarity statement and the scale factor. Then find x, y, and the measure of VU. [Image of two pentagons labeled ABCDE and RSTUV with side lengths labeled] Answer In the pentagon ABCDE and pentagon RSTUV ∠A = ∠R ∠B = ∠S ∠C = ∠T ∠D = ∠U ∠E = ∠V this implies that pentagon ABCDE ~ pentagon RSTUV now we have to find scale factor scale factor = CD/TU = x/3 3/2 = x/3 ⇒ 9 = 2x ⇒ x = 9/2 this is the required answer. and scale factor = DE/VU 3/2 = 8/(y+1) ⇒ 3(y+1) = 16 ⇒ 3y + 3 = 16 ⇒ 3y = 16 - 3 = 13 ⇒ y= 13/3 now, we have to find the value of VU VU = y + 1 = 13/3 + 1 = 13/3 + 3/3 = 16/3 this is the required answer. Question 2 Complete the similarity statement. [Image of two triangles labeled LMN and STR with side lengths labeled] △LMN ~ △? Given, In △MNL ML=9 NL=15 MN=12 In △STR SR=6 ST=8 RT=10 We have to complete the similarity statement △LMN ~ △? Answer If we take the ratio ML/SR = 9/6 = 3/2 NL/RT = 15/10 = 3/2 MN/ST = 12/8 = 3/2 So, ML/SR = NL/RT = MN/ST Then, △LMN ~ △RST Question 3 State the triangle similarity theorem that proves the triangles are similar. [Image of two triangles with side lengths labeled] Answer Step 1 In geometry, two shapes are similar if they are the same shape but different sizes. Similar triangles are easy to identify because you can apply three theorems specific to triangles. These three theorems, known as Angle - Angle (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS). Angle-Angle (AA) says that two triangles are similar if they have two pairs of corresponding angles that are congruent. The two triangles could go on to be more than similar. they could be identical. For AA, all you have to do is compare two pairs of corresponding angles. Step 2 SAS (Side-Angle-Side) If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, and then all three vertices of that triangle are fixed. With all three vertices fixed and two of the pairs of side's proportional, the third pair of sides must also be proportional. SSS (Side-Side-Side) Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar. Step 3 In the given figure, Both triangles are similar by SSS theorem, because all three pairs are in proportion, To calculate the proportion of sides, 15/10 = 3/2 24/16 = 3/2 12.8/8 = 1.6 Hence the proportion of sides is (1.6=8/5) therefore both triangles are similar. Question 4 Are the triangle similar, if so write a similarity statement. [Image of two triangles labeled TWZ and XWY with side lengths and angle measures labeled] Answer Step 1 If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar. Step 2 Now, In △TWZ and △XWY ∠TWZ = ∠XWY TW/XW = 10/15 = 2/3 ZW/YW = 11/20 TW/XW ≠ ZW/YW Therefore, these triangles are not similar. Question 5 Write the triangle similarity statement and the reason for why the triangles are similar. If they are not write the reason why. [Image of two triangles labeled MNQ and PNO with side lengths labeled and an angle marked.] △MNQ ~ △? By: ? What is the scale factor?? Answer Step 1 SAS or side angle side similiarty: If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar. We can see here ∠MNQ = ∠PNO (vertical angle) And MN/PN = 6/10 QN/ON = 6/10 = 5/10 Therefore, by the property SAS similiarty rule we can say △MNQ ~ △PNO BY SAS Step 2 Here, sale factor is 1/2.