Construction of a triangle given its perimeter and its two base angles

Construction of Triangle Given Its Perimeter and Its Two Base Angles:  In order to construct a triangle of given perimeter and two base angles, we follow the following steps:

Steps of Construction:  Obtain the perimeter and the base angles of the triangle.Let ABC be a triangle of perimeter p cm and base BC.

STEP I  Draw a line segment XY equal to the perimeter p of . STEP II Construct   and
STEP III : Draw bisectors of angles and and mark their intersection point as A.
STEP IV:  Draw the perpendicular bisectors PQ, RS  of XA and YA meeting XY in B and C respectively.
STEP V Join AB and AC to obtain the required triangle ABC.

Justification: For the justification of the construction, we observe that B lies on the perpendicular bisector of AX.

         XB = AB              [ Angles opposite to equal sides are equal]

Similarly, C lies on the perpendicular bisector of AY.

       YC = AC        

Now,  XY = XB + BC + CY

    XY = AB + BC + AC

In we

                   [Exterior angle is equal to the sum of interior opposite angle]

                                         [because the angles are equal Proved above]

                                             [ AX is the bisector ]

                            =                       [By Step II of Construction]

In we have

                            [Exterior angle is equal to the sum of interior opposite angle]

                                                   [because the angles are equal Proved above]

                                                   [ AY is the bisector ]

                                                               [By Step II of Construction]

Example: Construct a triangle PQR whose perimeter is equal to 14 cm, and .

SOLUTION  To draw we follow the following steps:

Steps of Construction:

STEP I  Draw a line segment XY = 14 cm STEP II Construct and
STEP III Draw the bisectors of angles and mark their point of intersection as R.
STEP IV  Draw right bisector AB, CD of RX and RY meeting XY at P and Q respectively.
STEP V Join PR and QR to obtain the required triangle PQR.