A phone company must place a new signal mast so that it is the same distance from Elm Street as it is from Oak Street, two straight roads meeting at a junct...
Question 1 Report
A phone company must place a new signal mast so that it is the same distance from Elm Street as it is from Oak Street, two straight roads meeting at a junction \(J\), as shown. Using ruler and compasses only, construct the locus of points where the mast could be placed. (2)
Answer Details
The locus of points equidistant from two straight roads meeting at a junction is the angle bisector of the angle between them at that junction, so the construction must produce this bisector using only compasses and a straightedge.
First, with compass point fixed at \(J\), draw a single arc of any radius that crosses both Elm Street and Oak Street; because it is one arc from one centre, both crossing points are the same distance from \(J\). [1 mark]
Then, keeping the compasses open to the same width (or any width more than half the distance between the two crossing points), draw an arc centred on each crossing point in turn, so that the two new arcs intersect. Every point on each of these arcs is equidistant from its own crossing point, and where they cross, that point is equidistant from both roads. Drawing a straight line from \(J\) through this intersection gives the required locus, the bisector of the angle at \(J\). [1 mark]
The construction works because the bisector of an angle is exactly the set of points equidistant from the two arms of that angle; no measuring of lengths or angles is needed, only equal compass radii.
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