Constructing similar triangle as per scale factor - Scale factor < 1

### Transcript

Example 2 Construct a triangle similar to a given triangle ABC with its sides equal its to 5/3 of the corresponding sides of the triangle ABC (scale factor 5/3). Here, we are given Δ ABC, and scale factor 5/3 ∴ Scale Factor > 1 We need to construct triangle similar to Δ ABC Let’s follow these steps Steps of construction Draw any ray BX making an acute angle with BC on the side opposite to the vertex A. Mark 5 (the greater of 5 and 3 in 5/3) points 𝐵_1,𝐵_2,𝐵_3,𝐵_4,𝐵_5on BX so that 〖𝐵𝐵〗_1=𝐵_1 𝐵_2=𝐵_2 𝐵_3=𝐵_3 𝐵_4 =𝐵_4 𝐵_5 Join 𝐵_3C (3rd point as 3 is smaller in 5/3) and draw a line through 𝐵_5 parallel to 𝐵_3 𝐶, to intersect BC extended at C′. Draw a line through C′ parallel to the line CA to intersect BA extendedat A′. Thus, Δ A′BC′ is the required triangle Justification Since scale factor is 5/3, we need to prove (𝑨^′ 𝑩)/𝑨𝑩=(𝑨^′ 𝑪^′)/𝑨𝑪=(𝑩𝑪^′)/𝑩𝑪 = 𝟓/𝟑. By construction, BC^′/𝐵𝐶=(𝐵𝐵_5)/(𝐵𝐵_3 )= 5/3. Also, A’C’ is parallel to AC So, they will make the same angle with line BC ∴ ∠ A’C’B = ∠ ACB Justification Since scale factor is 5/3, we need to prove (𝑨^′ 𝑩)/𝑨𝑩=(𝑨^′ 𝑪^′)/𝑨𝑪=(𝑩𝑪^′)/𝑩𝑪 = 𝟓/𝟑. By construction, BC^′/𝐵𝐶=(𝐵𝐵_5)/(𝐵𝐵_3 )= 5/3. Also, A’C’ is parallel to AC So, they will make the same angle with line BC ∴ ∠ A’C’B = ∠ ACB (Corresponding angles) Now, In Δ A’BC’ and Δ ABC ∠ B = ∠ B (Common) ∠ A’C’B = ∠ ACB (From (2)) Δ A’BC’ ∼ Δ ABC (AA Similarity) Since corresponding sides of similar triangles are in the same ratio (𝐴^′ 𝐵)/𝐴𝐵=(𝐴^′ 𝐶^′)/𝐴𝐶=(𝐵𝐶^′)/𝐵𝐶 So, (𝑨^′ 𝑩)/𝑨𝑩=(𝑨^′ 𝑪^′)/𝑨𝑪=(𝑩𝑪^′)/𝑩𝑪 = 𝟓/𝟑. (From (1)) Thus, our construction is justified

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.