Constructions, loci, bearings, plans and elevations – Year 11 GCSE maths revision notes to download as PDF

Worked example 1: region from loci

Find the area of points within 2 cm of a line segment of length 5 cm.

The region is a rectangle \(5\times4=20\ \text{cm}^2\) plus two semicircles that make one full circle of radius 2: \(\pi\times2^2=12.57\). Total \(=32.57\ \text{cm}^2\).

Worked example 2: a ship's journey

A ship sails 12 km from \(P\) to \(Q\) on a bearing of \(060^\circ\), then 5 km from \(Q\) to \(R\) on a bearing of \(150^\circ\). Find \(PR\) and the bearing of \(R\) from \(P\).

The back bearing from \(Q\) to \(P\) is \(060+180=240^\circ\). The angle between \(QP\) (\(240^\circ\)) and \(QR\) (\(150^\circ\)) is \(240-150=90^\circ\). So triangle \(PQR\) is right-angled at \(Q\) and \(PR=\sqrt{12^2+5^2}=13\) km. The bearing of \(R\) from \(P\) is \(060^\circ+\tan^{-1}\dfrac{5}{12}\approx082.6^\circ\).

Worked example 3: a map path

On a \(1:25\,000\) map, a path is 14 cm long. Find its real length.

Real length \(=14\times25\,000=350\,000\) cm \(=3.5\) km.

Worked example 4: views of solids

Give the plan and elevations of a cuboid \(4\ \text{cm}\times3\ \text{cm}\times2\ \text{cm}\) high, a cone and a triangular prism lying on a rectangular face.

The cuboid has a plan \(4\times3\), a front elevation \(4\times2\) and a side elevation \(3\times2\). A cone has a circular plan and a triangular elevation. The prism has a rectangular plan and one rectangular elevation, and its other elevation is a triangle.

Worked example 5: Back bearings

The bearing of \(P\) from \(Q\) is \(305^\circ\). Find the bearing of \(Q\) from \(P\).

\(305\) is greater than \(180\), so subtract \(180\): \(305-180=125^\circ\). Check by drawing: from \(Q\) the point \(P\) is north-west, so from \(P\) the point \(Q\) is south-east, which is a bearing near \(135^\circ\). The answer \(125^\circ\) is in the correct quadrant. Always sketch a north line at each point.

Worked example 6: Converting a scale

A model of a ship is built at a scale of \(1:80\). The model is 45 cm long. Find the length of the real ship in metres.

\(45\times80=3600\) cm. Divide by 100 to change to metres: \(3600\div100=36\) m. Always convert to a convenient unit at the end. For areas the scale factor is squared: \(1\) cm\(^2\) on the model is \(80^2=6400\) cm\(^2\) on the ship.

Worked example 7: Constructing a \(60^\circ\) angle

Describe how to construct a \(60^\circ\) angle at point \(A\) on a line using only compasses and a ruler.

Place the compass point at \(A\) and draw an arc across the line at \(B\). Keep the same radius, put the point at \(B\) and draw an arc crossing the first arc at \(C\). Join \(A\) to \(C\). Triangle \(ABC\) has three equal sides, so it is equilateral and angle \(BAC=60^\circ\). Bisecting it gives \(30^\circ\).

Worked example 8: A locus of equal distance from two lines

Lines \(AB\) and \(AC\) meet at \(A\) with angle \(BAC=50^\circ\). Describe the locus of points equidistant from the two lines and state the angle it makes with each.

The locus is the angle bisector of \(BAC\). To draw it, put the compass point at \(A\) and draw an arc across both lines. From the two crossing points draw arcs of the same radius that meet. Draw a line from \(A\) through the meeting point. It makes \(25^\circ\) with each line. Every point on it is the same perpendicular distance from the two lines.

Worked example 9: Using the plan and elevations of a solid

The plan of a solid is a circle of diameter 6 cm, the front elevation is a rectangle 6 cm by 10 cm. Name the solid and find its volume.

A circle from above and a rectangle from the front mean a cylinder of diameter 6 cm and height 10 cm. Radius \(=3\) cm. Volume \(=\pi r^2h=\pi\times3^2\times10=90\pi=282.7\) cm\(^3\). If the front elevation were a triangle, the solid would be a cone.

Worked example 10: A bearing in a triangle of sides

A walker goes 5 km due east then 12 km due north. Find the bearing of the start from the walker.

The final position is \((5,12)\). The bearing of the walker from the start is \(\tan^{-1}(5/12)=22.6^\circ\), that is \(023^\circ\). The bearing of the start from the walker is the back bearing \(22.6+180=202.6\approx203^\circ\). The distance is \(\sqrt{5^2+12^2}=13\) km.

Worked example 11: Area on a scale drawing

On a plan with scale \(1:200\), a rectangular car park measures 4 cm by 3 cm. Find the real area in square metres.

The real sides are \(4\times200=800\) cm \(=8\) m and \(3\times200=600\) cm \(=6\) m. Area \(=8\times6=48\) m\(^2\). Alternatively the plan area is 12 cm\(^2\) and the scale factor for area is \(200^2=40\,000\), giving \(480\,000\) cm\(^2=48\) m\(^2\).