Question

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.

Answer 1

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.

Justification:

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`