Question

Draw two intersecting lines to include an angle of 30${}^{\circ }$. Use ruler and compass to locate points which are equidistant from these lines and also 2 cm away from their point of intersection. How many such point/points exist(s)?

Answer 1

Draw two lines AB and CD such that they intersect at O and $\angle$AOC = 30${}^{\circ }$.

We know that the locus of a point which is equidistant from two intersecting straight lines is a pair of straight lines which bisect the angles between the given lines.

So, points equidistant from these lines lie on the bisectors of $\angle$AOC, $\angle$AOD, $\angle$DOB and $\angle$BOC.

Therefore we draw the bisectors of these angles.

Now, draw a circle with O as centre and radius 2cm.

As we see from the figure above, his circle cuts OP, OR, OQ and OS at four points L, M, N and U respectively.

Hence, there are four points which are equidistant from AB and CD and 2 cm away from O.