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.

#### Solution

**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 B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6} and B_{7} on AX, such that AB_{1} = B_{1}B_{2} = B_{2}B_{3} = B_{3}B_{4} = B_{4}B_{5} = B_{5}B_{6}= B_{6}B_{7}.

6. Join B_{7}B and from B_{5} draw B_{5}M ॥ B_{7}B 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.

Justification:

Here, B_{5}M ॥ B_{7}B .....(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`