Construct the following and give justification :

A triangle if its perimeter is 10.4 cm and two angles are 45^{\circ} and 120^{\circ}.

Answers (1)

Steps of Construction :

  1. Draw a line PQ = 10.4 cm
  2. Draw line PR with 45^{\circ} angle at P.

\angle RPQ=45^{\circ}

  1. Draw line QS with 120^{\circ} angle at Q.

                     \angle SPQ=120^{\circ}

            3.         Bisect \angle RPQ and \angle SPQ and then let their bisectors meet at point A

            4.         Bisect line AQ and AP which meets PQ at C and B respectively.

            5.         Join A with B and C.

            ABC is the required triangle.

           

            Justification:

            Since B and C lies on PQ

            Hence, PB + BC + CQ = AB + BC + AC = PQ

                     \angle BAP= \angle APB  …..(i)                    { in DAPB, AB = PB}

                      \angle ABC= \angle BPA+ \angle APB             [\angle ABC is an exterior angle of DAPB]

                      \angle ABC= 2 \angle APB = \angle RPQ          {from (i),\angle RPQ= \angle RPA + \angle APQ}

            Also,  \angle CAQ = \angle AQC                        [in DAQC ; AC = CQ]

                       \angle ACB = \angle CAQ + \angle AQC            [\angle ACB is exterior angle of DAQC]

                      \angle ACB = 2 \angle CAQ = \angle SQP         [\AQ is a bisector of \angle SQP]

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