Theory question 111
LociOn a graph sheet, using a scale of 2cm to 2 units on both axes. (a) draw the straight line joining points P (−5,3) and Q (2,3); (b) construct the locus L of points equidistant from P and Q; (c) by construction, locate points R and S on L, such that PRQS form a rhombus of sides 5cm; (d) find; (i) coordinates of R and S; (ii) area of the rhombus in cm2.
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(a) PQ is horizontal, length 7 units.
(b) L is the perpendicular bisector of PQ: the line x = −1.5.
(c) With centres P and Q and radius 5 units, draw arcs cutting L at R (above) and S (below).
(d)(i) Vertical distance = √(52 − 3.52) = √12.75 ≈ 3.57, so R ≈ (−1.5, 6.6) and S ≈ (−1.5, −0.6).
(ii) Area = ½ × PQ × RS = ½ × 7 × 7.14 ≈ 25.0 cm2 (1 unit = 1 cm on this scale).
Every point of a rhombus diagonal's perpendicular bisector is equidistant from its ends, so R and S lie on L.