Question

Ruler and compasses may be used in this question. All construction lines and arcs must be clearly shown and be of sufficient length and clarity to permit assessment.

1. Construct a ΔABC, in which BC = 6 cm, AB = 9 cm and angle ABC = 60°.
2. Construct the locus of all points inside triangle ABC, which are equidistant from B and C.
3. Construct the locus of the vertices of the triangles with BC as base and which are equal in area to triangle ABC.
4. Mark the point Q, in your construction, which would make ΔQBC equal in area to ΔABC, and isosceles.
5. Measure and record the length of CQ.

Answer 1

 
Steps of construction:

1. Draw a line segment BC = 6 cm.
2. At B, draw a ray BX making an angle 60 degree and cut off BA = 9 cm.
3. Join AC. ABC is the required triangle.
4. Draw perpendicular bisector of BC which intersects BA in M, then any point on LM is equidistant from B and C.
5. Through A, draw a line m || BC.
6. The perpendicular bisector of BC and the parallel line m intersect each other at Q.
7. Then triangle QBC is equal in area to triangle ABC. m is the locus of all points through which any triangle with base BC will be equal in area of triangle ABC.
   On measuring CQ = 8.4 cm.