Exercise solution 4, page 89 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) An angle is formed by two rays meeting at one point, called the vertex. In \(\angle MNO\), the middle letter \(N\) is the vertex, so the angle is made by the rays \(NM\) and \(NO\). Saying \(\angle MNO = 50^\circ\) means the opening between these two rays is \(50^\circ\).

B) To construct an angle accurately, you usually use a ruler and a protractor. First draw one side of the angle, place the centre of the protractor on the vertex, line up \(0^\circ\) with the side already drawn, mark the required angle, then draw the second side through that mark. For example, to construct an angle of \(40^\circ\), you would mark \(40^\circ\) from the starting ray and join that mark to the vertex.

C) A locus is the set of all points that satisfy a given condition. For example, the locus of points that are exactly \(4\text{ cm}\) from a fixed point is a circle of radius \(4\text{ cm}\), centred at that point. In locus questions, the important idea is to identify all points with the required property, not just one point.

D) The distance from a point to a line is measured along the shortest path, which is a perpendicular line segment. So if a point is equidistant from two lines, its perpendicular distance to the first line is equal to its perpendicular distance to the second line.

E) The locus of points equidistant from two intersecting lines is the angle bisector. An angle bisector is a line that divides an angle into two equal angles. For instance, if two lines form an angle of \(70^\circ\), the angle bisector splits it into two angles of \(35^\circ\).

F) In a diagram with rays \(AB\) and \(AC\), if the angle \(\angle BAC\) is \(80^\circ\), then the bisector creates two equal angles:

\[\begin{aligned} \angle BAD &= \angle DAC \\ &= \frac{80^\circ}{2} \\ &= 40^\circ \end{aligned}\]

This is the key idea when the question asks for points equidistant from the two sides of an angle.

G) When constructing an angle bisector with compasses, you first draw an arc from the vertex cutting both sides of the angle. Then, using the same compass width, draw arcs from those two intersection points so that the arcs cross inside the angle. Joining the vertex to the crossing point gives the angle bisector. This line represents the locus of points equidistant from the two sides of the angle.

H) In geometry constructions, accuracy matters. Use a sharp pencil, keep the compass width fixed when needed, and draw arcs clearly enough to show how the construction was made. A correct locus should be drawn with a ruler only after the necessary compass arcs have been constructed.