Question

Draw a line segment AB of length 12 cm. Mark M, the midpoint of AB. Draw and describe the locus of a point which is

1. at a distance of 3 cm from AВ. 
2. at a distance of 5 cm from the point М. 

Mark the points P, Q, R, S which satisfy both the above conditions. What kind of quadrilateral is PQRS? Compute the area of the quadrilateral PQRS.

Answer 1

1. The locus of a point 3 cm from AB is two lines parallel to AB at a distance 3 cm.
2. The locus of a point 5 cm from M is a circle with centre M and radius 5 cm.

P, Q, R, and S are the four intersection points of this circle with the two parallel lines. PQ ∥ RS and PR ∥ QS, and all angles are right angles, so PQRS is a rectangle. MQ = MP = 5 cm and MB = 0, AB = 12 cm ⇒ PQ = RS = 8 cm, PR = QS = 6 cm,
So area PQRS = 8 × 6 = 48 cm².