Exercise solution 8, page 89 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) A locus is the set of all points that satisfy a given condition. In this type of question, you are not looking for just one point, but for every point that fits the rule. For example, the locus of points that are 2 cm from a fixed point is a circle of radius 2 cm centred on that point.

B) The distance from a point to a side of a square is measured by the shortest distance, which is a line drawn perpendicular to that side. So when a question says that a point is less than a certain distance from a side, you must think about perpendicular distance, not sloping distance.

C) The locus of points at a fixed distance from a straight line is made of two lines parallel to the original line, one on each side. For instance, the points that are exactly 3 cm from a line form two parallel lines, each 3 cm away. If the condition is less than 3 cm, then the locus is the strip of points between those two parallel lines.

D) For a square, you need to consider each side separately. The points less than a given distance from one side form a strip next to that side. Repeating this for all four sides helps you identify the region that satisfies the condition. The required locus is usually the part where these strips combine according to the wording of the question.

E) The phrase less than means the boundary itself is not included. So if a point must be less than 1 cm from a side, points exactly 1 cm away do not belong to the locus. On a diagram, this often means the boundary line is a guide for construction, but it is not part of the final locus. Remember that \(d < 1\) is different from \(d \leq 1\).

F) In a construction question, accuracy matters. You usually draw lines parallel to the sides of the square at the required distance using a ruler and compass or careful measuring. The final locus is the region shown by your construction, so it should be shaded or indicated clearly.

G) It is useful to think about what happens near the corners. A point near a corner can be close to two sides at the same time. This helps you understand the shape of the region you are constructing. Always check whether the condition refers to being close to the sides of the square, since that means all parts of the diagram must be judged using distance to the sides, not distance to the vertices.