Theory question 110
ConstructionsUsing ruler and a pair of compasses only, (a) construct: (i) triangle XYZ with |XY| = 8cm, ∠YXZ = 60° and ∠XYZ = 30°; (ii) the perpendicular ZT to meet XY in T; (iii) the locus l1 of points equidistant from ZY and XY. (b) If l1 and ZT intersect at S, measure |ST|.
Show the model answer
Model answer
(a)(i) Draw XY = 8 cm. Construct 60° at X and 30° at Y (bisect a 60° angle); the arms meet at Z (∠Z = 90°).
(ii) From Z drop the perpendicular to XY, meeting it at T.
(iii) Bisect ∠XYZ; this bisector is l1.
(b) ST ≈ 1.6 cm (by calculation: XZ = 4, XT = 2, TY = 6, ST = 6 tan 15° ≈ 1.61 cm).
Explanation
Points equidistant from two lines lie on the bisector of the angle between them.