(a) Using a ruler and a pair of compasses only, construct (i) a triangle XYZ in which /YZ/ = 8cm, < XYZ = 60° and < XZY = 75°. Measure /XY/; (ii) the locus ...
Question 1 Report
(a) Using a ruler and a pair of compasses only, construct (i) a triangle XYZ in which /YZ/ = 8cm, < XYZ = 60° and < XZY = 75°. Measure /XY/; (ii) the locus \(l_{1}\) of points equidistant from Y and Z ; (iii) the locus \(l_{2}\) of points equidistant from XY and YZ.
(b) Measure QY where Q is the point of intersection of \(l_{1}\) and \(l_{2}\).
Answer Details
(a)(i) Construction of triangle \(XYZ\)
- Draw \(YZ=8\text{ cm}\).
- At \(Y\), construct an angle of \(60^\circ\). At \(Z\), construct an angle of \(75^\circ\), using ruler and compasses only.
- Let the two constructed rays meet at \(X\). This gives triangle \(XYZ\).
From the completed construction,
\[|XY|\approx 10.9\text{ cm}.\]
(a)(ii) The locus \(l_1\) of points equidistant from \(Y\) and \(Z\) is the perpendicular bisector of \(YZ\).
(a)(iii) The locus \(l_2\) of points equidistant from \(XY\) and \(YZ\) is the internal bisector of \(\angle XYZ\).
(b) Let \(Q\) be the intersection of \(l_1\) and \(l_2\). Measuring from the construction gives
\[\boxed{|QY|\approx 4.8\text{ cm}}\]
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