# Draw a triangle ABC in which AB = 4 cm, BC = 6 cm and AC = 9 cm. Construct a triangle similar to $\bigtriangleup ABC$ 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

Solution

Steps of construction

1. Draw a line segment BC = 6 cm
2. Taking B and C as a centres, draw arc of radius AB= 4cm and AC=9cm
3. Join AB and AC
4. DABC is required triangle From B draw ray BM with acute angle $\angle XBM$
5. Make 3 points $B_{1},B_{2},B_{3}$  on BM with equal distance
6. Join $B_{2}C$  and $B_{3}$  draw $B_{3}X\parallel B_{2}C$  intersecting BC at X From point X draw XY||CA intersecting the extended line segment BA to Y Then $\bigtriangleup BXY$  is the required triangle whose sides are equal to$\frac{3}{2}$  of the $\bigtriangleup ABC$
Justification :
Here $B_{3}X\parallel B_{2}C$
$\therefore \frac{BC}{CX}= \frac{2}{1}$
$\therefore \frac{BX}{BC}= \frac{BC+CX}{BC}= 1+\frac{1}{2}= \frac{3}{2}$
Also $XY\parallel CA$
$\bigtriangleup ABC\sim \bigtriangleup YBX$
$\therefore \frac{YB}{AB}= \frac{YX}{AC}= \frac{BX}{BC}= \frac{3}{2}$
Here all the three angles are same but three sides are not same.
$\therefore$The two triangles are not congruent because, if two triangles are congruent, then they have same shape and same size.

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