Loci constructions and bearings
Geometry and measures
- Crossover
- Calculator allowed
- 8 questions
- 29 marks
Questions marked Higher are aimed at the Higher tier.
These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.
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Write the three-figure bearing of a point that is due east of the start.
2 marks
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A point is on a bearing of $200°$. Find the bearing looking back from that point.
2 marks
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From $A$, point $B$ is on a bearing of $060°$. Write the bearing of $A$ from $B$.
3 marks
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Describe the locus of points that are $3$ cm from a fixed point $A$, and the locus of points equidistant from two fixed points $A$ and $B$.
3 marks
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A treasure is inside a circle of radius $5$ m around a tree, and more than $3$ m from a straight wall. Describe the region. A dog is tied to a corner of an $8$ m by $6$ m rectangular garden by a $5$ m rope. Find the area the dog can reach inside the garden, in terms of $\pi$.
4 marks
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A plane flies $40$ km due north and then $30$ km due east. Find the distance from the start, and the bearing of the plane from the start to the nearest degree.
4 marks
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A rectangular garden on a worksheet is $8$ cm by $6$ cm. A sprinkler at one corner waters the region within $5$ cm. Describe that region inside the rectangle. Also describe the path of points equidistant from the two long sides. Higher
5 marks
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From $A$, a boat sails $12$ km due east to $B$ and then $9$ km due south to $C$, as shown. Find the distance $AC$, the bearing of $C$ from $A$ to the nearest degree, and the bearing of $A$ from $C$. Describe the locus of points equidistant from $A$ and $C$. Higher
Journey 12 km east from A to B, then 9 km south to C 6 marks
Mark scheme
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Answer. $090°$
- East is a quarter turn clockwise from north.
- $090°$
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Answer. $020°$
- Add or subtract $180°$.
- $200 - 180 = 020°$
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Answer. $240°$
- The reverse bearing is $60 + 180$.
- $240°$
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Answer. A circle, centre $A$, radius $3$ cm. The perpendicular bisector of $AB$.
- A fixed distance from a point gives a circle.
- Equal distance from two points gives the perpendicular bisector.
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Answer. Inside the circle and beyond the line $3$ m from the wall. The dog can cover a quarter circle of area $\dfrac{25\pi}{4}$ m$^{2}$.
- The treasure region is the part of the circle on the far side of the line parallel to the wall.
- The rope is shorter than both sides, so the dog covers a quarter circle.
- Area $= \dfrac{1}{4}\pi \times 5^{2} = \dfrac{25\pi}{4}$ m$^{2}$.
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Answer. $50$ km, on a bearing of $037°$
- Distance $= \sqrt{30^{2} + 40^{2}} = 50$ km.
- $\tan \theta = \dfrac{30}{40} = 0.75$, so $\theta = 36.9°$ to $1$ d.p.
- The bearing is $037°$.
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Answer. A quarter circle of radius $5$ cm fits inside the rectangle. The path is the centre line, parallel to the long sides and $3$ cm from each.
- $5$ cm is less than both $6$ cm and $8$ cm, so the quarter circle stays inside.
- The set of points equidistant from two parallel sides is the line halfway between them.
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Answer. $AC = 15$ km. Bearing of $C$ from $A$ is $127°$. Bearing of $A$ from $C$ is $307°$. The locus is the perpendicular bisector of $AC$.
- $AC = \sqrt{12^{2} + 9^{2}} = 15$ km.
- $\tan \theta = \dfrac{9}{12}$, so the angle south of east is $36.9°$. The bearing is $090 + 37 = 127°$.
- The back bearing is $127 + 180 = 307°$.
- Points equidistant from $A$ and $C$ lie on the perpendicular bisector of $AC$.