Question

Draw a $\Delta ABC$ in which AB= 4 cm, BC = 6 cm and AC = 9 cm, construct a triangle similar to $\Delta ABC$ with scale factor $\frac{3}{2}$. 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

Thinking process

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.

So, first, we construct a triangle similar to $\Delta ABC$ with scale factor $\frac{3}{2}$ 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 centers. 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 $B{B}_1=B_1{B}_2=B_2{B}_3$
6. Join $B_{2}C$ and from $B_3$ draw $B_{3}M\parallel B_{2}C$ intersecting the extended line segment BC at M
7.From point M, draw $MN\parallel CA$ intersecting the extended line segment BA to N then, $\Delta NBM$ is the required triangle whose sides are equal to $\frac{3}{2}$ of the corresponding sides of the $\Delta ABC$.

Justification.

Here, $B_{3}M\parallel B_{2}C$

$\therefore \frac{BM}{BC}=\frac{2}{1}$

Now, $\frac{BM}{BC}=\frac{BC+CM}{BC}$

$=1+\frac{CM}{BC}=1+\frac{1}{2}=\frac{3}{2}$

Also $MN\parallel CA$

$\therefore \Delta ABC\sim \Delta NBM$

Therefore, $\frac{NB}{AB}=\frac{NM}{AC}=\frac{BM}{BC}=\frac{3}{2}$

The two triangles are not congruent because if two triangles are congruent, then they have the same shape and same size, here all the three angles are same but three sides are not same .