Question

Draw a triangle ABC in which AB = 4 cm, BC = 6 cm and AC = 9 cm. Construct a triangle similar to ∆ABC with scale factor `3/2`. Justify the construction. Are the two triangles congruent? Note that all the three angles and two sides of the two triangles are equal.

Answer 1

Steps of construction:

1. Draw a line segment BC = 6 cm.
2. Taking B and C as centres, draw two arcs of radii 4 cm and 9 cm intersecting each other at A.
3. Join BA and CA, ΔABC is the required triangle.
4. From B, draw any ray BX downwards making an acute angle.
5. Mark three points B₁, B₂, B₃ on BX, such that BB₁ = B₁B₂ = B₂B₃.
6. Join B₂C and from B₃ draw B₃M || B₂C intersecting the extended line segment BC at M.
7. From point M, draw MN || CA intersecting the extended line segment BA to N.
   Then, ΔNBM is the required triangle whose sides are equals to `3/2` of the corresponding sides of the ΔABC.

Justification:

Here, B₃M || B₂C

∴ `"BC"/"CM" = 2/1`

Now, `"BM"/"BC" = ("BC" + "CM")/"BC"`

= `1 + "CM"/"BC"`

= `1 + 1/2`

= `3/2`

Also, MN || CA

∴ ΔABC ∼ ΔNBM

Therefore, `"NB"/"AB" = "NM"/"AC" = "BM"/"BC" = 3/2`

The two triangles are not congruent because, if two triangles are congruent, then they have same shape and same size. Here, all the three angles are same but three sides are not same i.e., one side is different.