Theory question 108
Constructions(a) Using a ruler and a pair of compasses only, construct: (i) a traingle XYZ in which |YZ| = 8cm, XYZ = 60° and XZY = 75°. Measure |XY|; (ii) the locus l1 of points equidistant from Y and Z; (iii) the locus l2 of points equidistant from YX and YZ.
(b) Measure QY where Q is the point of intersection of l1 and l2
Show the model answer
Model answer
(a)(i) Draw YZ = 8 cm; construct 60° at Y and 75° at Z (60° + bisected 30°, i.e. 60° + 15°); the arms meet at X.
|XY| ≈ 10.9 cm (check: 8 sin 75°/sin 45° = 10.93)
(ii) l1 is the perpendicular bisector of YZ.
(iii) l2 is the bisector of angle XYZ.
(b) Q lies on the bisector (30° to YZ) above the midpoint of YZ: QY ≈ 4.6 cm (4/cos 30° = 4.62)
Explanation
Leave all construction arcs visible; a locus equidistant from two lines is an angle bisector.