Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60°. Construct a triangle similar to ∆ABC with scale factor `5/7`. Justify the construction.
Steps of construction:
1. Draw a line segment AB = 5 cm.
2. From point B, draw ∠ABY = 60° on which take BC = 6 cm.
3. Join AC, ∆ABC is the required triangle.
4. From A, draw any ray AX downwards making an acute angle.
5. Mark 7 points B1, B2, B3, B4, B5, B6 and B7 on AX, such that AB1 = B1B2 = B2B3 = B3B4 = B4B5 = B5B6= B6B7.
6. Join B7B and from B5 draw B5M ॥ B7B intersecting AB at M.
7. From point M draw MN ॥ BC intersecting AC at N. Then, ∆AMN is the required triangle whose sides are equal to `5/7` of the corresponding sides of the ∆ABC.
Here, B5M ॥ B7B .....(By construction)
∴ `(AM)/(MB) = 5/2`
Now, `(AB)/(AM) = (AM + MB)/(AM)`
= `1 + (MB)/(AM) = 1 + 2/5 = 7/5`
Also, MN || BC
∴ ∆AMN ∼ ∆ABC
Therefore, `(AM)/(AB) = (AN)/(AC) = (NM)/(BC) = 5/7`