Theory question 107
Constructions(a) Using a ruler and a pair of compasses only, construct ΔABC in which |AB| = 7cm, |BC| = 5cm and ∠ABC = 75°. Measure AC.
(b) In (a) above, locate by construction, a point D such that CD is parallel to AB and D is equidistant from points A and C. Measure ∠BAD.
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(a) Draw BC = 5 cm. At B construct 75° (60° plus half of the 30° between 60° and 90°). Mark A on this arm with BA = 7 cm and join AC.
AC ≈ 7.5 cm (by calculation √(49 + 25 − 70 cos 75°) ≈ 7.48 cm).
(b) Through C construct a line parallel to AB (copy ∠ABC at C, or use parallelogram construction). Construct the perpendicular bisector of AC; D is where it meets the parallel line.
∠BAD ≈ 80° (accept 79°–81°).
D must lie on two loci at once: the line through C parallel to AB and the perpendicular bisector of AC.