Exercise solution 7, page 89 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) A rectangle has four right angles, and opposite sides are equal and parallel. If you are asked to draw a rectangle from two side lengths, you should use a ruler for the lengths and make sure the corners are \(90^\circ\). For example, a rectangle with sides \(8\text{ cm}\) and \(5\text{ cm}\) has two long sides of \(8\text{ cm}\) and two short sides of \(5\text{ cm}\).

B) A locus is the set of all points that satisfy a given rule. In geometry, a locus is often a line, a curve, or a shaded region. For example, the locus of points that are \(3\text{ cm}\) from a fixed point is a circle of radius \(3\text{ cm}\), centred at that point.

C) Points that are equidistant from two intersecting lines lie on the angle bisectors of the angles formed by those lines. An angle bisector divides an angle into two equal parts. So, if two sides of a shape meet at a corner, the points equally far from both sides lie on the line that bisects that corner angle. In a right angle, this bisector makes angles of \(45^\circ\) with each side.

D) The distance from a point to a line is measured by the shortest distance, which is the perpendicular distance. This means you do not measure diagonally to the line unless that diagonal is perpendicular. When a question says a point is a certain distance from a side, think of the distance measured at right angles to that side.

E) The centre of a rectangle is the point where its diagonals cross. A rectangle has two diagonals, and they bisect each other, so their intersection gives the centre exactly. This point is equally far from opposite corners.

F) The locus of points that are less than a given distance from a fixed point is the inside of a circle. The fixed point is the centre, and the given distance is the radius. For instance, the set of points less than \(2.5\text{ cm}\) from a point \(P\) is the region inside the circle with centre \(P\) and radius \(2.5\text{ cm}\). Because the wording says “less than”, the region is shaded inside the boundary.

G) The locus of points that are exactly a given distance from a fixed point is the circumference of a circle. The word “exactly” means only the boundary, not the inside. So it is important to notice the difference between “less than”, “more than”, and “exactly”.

H) The locus of points that are a fixed distance from the perimeter of a shape can be drawn by imagining a line all the way around the shape at that constant distance. For a rectangle, this gives straight segments parallel to each side, joined by curved parts around the corners. Around each corner, the curved part is a quarter-circle whose radius is the given distance.

I) If the locus is outside the rectangle, you must draw it beyond the edges of the shape, not inside it. For example, points exactly \(1.5\text{ cm}\) outside a rectangle would be found on lines parallel to each side at a distance of \(1.5\text{ cm}\), together with quarter-circles of radius \(1.5\text{ cm}\) centred at the corners.

J) In construction questions, neatness matters. Use a ruler for straight lines, a compass for circles or arcs, and label important points clearly. If a region is required, shade the correct area lightly so that the construction lines can still be seen.