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

Answers (1)

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.
    \thereforeThe two triangles are not congruent because, if two triangles are congruent, then they have same shape and same size.

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