Draw a ∆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.

Thinking process:

I. Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.

II. So, first we construct a triangle a triangle similar to ∆ABC with scale factor 3/2 and use the above concept to check the triangles are congruent or not.

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, ∆ABC 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.

8. Then, ∆NBM is the required triangle whose sides are equal to the corresponding sides of the ∆ABC.

Justification:

Let BB₁= B₁B₂ = B₂B₃ = x

As per the construction B₃M││B₂C

Now,

Also from the construction MN||CA

_{Therefore rABC is congruent to rNBM}

_{(by the AA criteria, as ∠ NBM = ∠ ABC , also as MN||CA, ∠ ACB is corresponding to ∠ NMB so ∠ ACB = ∠ NMB)}

and

Hence, the new triangle rNBM is similar to the given triangle rABC and its sides are times of the corresponding sides of rABC.

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.