WAEC Mathematics 2001 Theory question 109

Theory question 109

Constructions

Using a ruler and a pair of compasses only, (a) Construct a quadrilateral MNOP with |OP| = 11 cm, |PM| = 6cm, |NO| = 5 cm, |MN| = 8 cm and |MO| = 9cm. (b) Measure ∠MPO. (c) Locate the point Y which is equidistant from the straight lines ON and OP and also equidistant from points M and N. (d) Draw YP and find the perimeter of ΔOYP.

Show the model answer

Model answer

(a) Draw OP = 11 cm. Arcs: centre P radius 6 cm and centre O radius 9 cm meet at M. Then arcs centre M radius 8 cm and centre O radius 5 cm meet at N (same side as M). Join MN, NO, PM.
(b) ∠MPO ≈ 55° (cos P = (36 + 121 − 81)/132).
(c) Bisect ∠NOP and construct the perpendicular bisector of MN; Y is where they cross.
(d) OY ≈ 5.2 cm, YP ≈ 8.4 cm, OP = 11 cm: perimeter ≈ 24.6 cm (accept 24–25 cm).

Explanation

Equidistant from two lines means on the angle bisector; equidistant from two points means on the perpendicular bisector.