Answer :

__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 B_{1},B_{2},B_{3} on BX, such that BB_{1}=B_{1}B_{2}=B_{2}B_{3}.

6. Join B_{2}C and from B_{3} draw B_{3}M││B_{2}C 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 BB_{1}= B_{1}B_{2} = B_{2}B_{3} = x

As per the construction B_{3}M││B_{2}C

Now,

Also from the construction MN||CA

_{Therefore} _{r}_{ABC is congruent to} _{r}_{NBM}

_{(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.

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