Question

Ruler and compasses only 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.
(i) Construct a ΔABC, in which BC = 6 cm, AB = 9 cm and ∠ABC = 60°.
(ii) Construct the locus of the vertices of the triangles with BC as base, which are equal in area to ΔABC.
(iii) Mark the point Q, in your construction, which would make ΔQBC equal in area to ΔABC, and isosceles.
(iv) Measure and record the length of CQ.

Answer 1

Steps of  Constructions:
(i) (1) Mark a horizontal line XY on your paper and take BC = 6 cm on it.
(2) Construct ∠ABC = 60° with arm AB = 9 cm.
(3) Join A and C to get the required ΔABC.
(ii) (1) Draw AD ⊥ BC.

(2) Construct a line X'Y', perpendicular to AD, parallel to XY and passing through A.
(3) X'Y', is the required locus of the vertices of Δ^s with base BC and area to ΔABC.
[∵ Δ^s having same base and height an equal in area]
(iii) (1) Draw right bisector PQ of BC, meeting X'Y', in Q.
(2) Then Q is the point such that ΔQBC is an isosceles triangle and area (ΔQBC) = area (ΔABC).
(iv) On measuring, we find  CQ = 8·4 cm.