Example 1
Construct a triangle similar to a given triangle ABC with its sides equal to 3/4 of the corresponding sides of the triangle ABC (i.e. of scale factor 3/4).
Here, we are given Δ ABC, and scale factor 3/4
∴ 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 4 (the greater of 3 and 4 in 3/4 ) points
𝐵_1,𝐵_2,𝐵_3 and 𝐵_4 on BX so that 〖𝐵𝐵〗_1=𝐵_1 𝐵_2=𝐵_2 𝐵_3=𝐵_3 𝐵_4.
Join 𝐵_4C
and draw a line through 𝐵_3 (the 3rd point, 3 being smaller of 3 and 4 in 3/4) parallel to 𝐵_4 𝐶,
to intersect BC at C′.
4. Draw a line through C′ parallel to the line CA to intersect BA at A′.
Thus, Δ A′BC′ is the required triangle
Justification
Since scale factor is 3/4,
we need to prove (𝑨^′ 𝑩)/𝑨𝑩=(𝑨^′ 𝑪^′)/𝑨𝑪=(𝑩𝑪^′)/𝑩𝑪 =𝟑/𝟒.
By construction,
BC^′/𝐵𝐶=(𝐵𝐵_3)/(𝐵𝐵_4 )=3/4.
Also, A’C’ is parallel to AC
So, they will make the same angle with line BC
∴ ∠ A’C’B = ∠ ACB
Now,
In Δ A’BC’ and Δ ABC
∠ B = ∠ B
∠ A’C’B = ∠ ACB
Δ A’BC’ ∼ Δ ABC
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 =𝟑/𝟒.
Thus, our construction is justified

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.