Maps Scale Drawings and Bearings
This topic covers the interpretation of maps and scale drawings, focusing on calculating real-world distances and using bearings for navigation. It tests your ability to apply geometric principles, particularly with angles and parallel lines, in practical scenarios without a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- A map scale, e.g., 1:50000, means 1 unit on the map represents 50000 of the same unit in reality. Ensure consistent units when calculating.
- Bearings are always measured clockwise from a North line.
- Bearings must always be written as a three-figure number, e.g., 035° for 35 degrees.
- All North lines drawn on a diagram are parallel to each other. This is the key to finding unknown angles using properties of parallel lines (e.g., co-interior angles sum to 180°).
- The bearing of point B *from* point A is measured at point A. Always draw the North line at the 'from' point.
Formulae
Back Bearing = Bearing ± 180 To find the bearing of A from B when you know the bearing of B from A. Add 180 if the original bearing is less than 180, subtract 180 if it is greater.
Definitions
- Scale
- The ratio that defines the relationship between a distance on a map or drawing and the corresponding actual distance.
- Three-Figure Bearing
- An angle, given as a 3-digit number from 000 to 359, measured clockwise from the North direction to specify a path or location.
Worked example
A town P is 12 km due south of a town Q. A third town, R, is on a bearing of 120° from Q and on a bearing of 060° from P. Without a calculator, find the distance in km between towns P and R.
- 1
First, sketch the points.
Draw Q, then draw P 12 km directly below it (due south).
Draw a North line at Q and at P (these lines are parallel).
- 2
From Q, draw a line at a 120° clockwise angle from North.
This is the direction to R.
- 3
From P, draw a line at a 060° clockwise angle from North.
The intersection of this line and the line from Q is town R.
- 4
You have formed a triangle PQR.
Now find the internal angles.
- 5
The line PQ is a North-South line.
At Q, the angle NQR is 120°.
The angle PQR inside the triangle is 180° - 120° = 60°.
- 6
At P, the angle NPR is 060°.
Since PQ is a North-South line, the angle QPR inside the triangle is also 60°.
- 7
Since two angles of the triangle are 60°, the third angle (PRQ) must also be 180° - 60° - 60° = 60°.
- 8
The triangle PQR is equilateral, meaning all its sides are equal in length.
- 9
The distance PQ is given as 12 km.
Therefore, the distance PR is also 12 km.
Answer: 12 km
Common mistakes
- ×Misinterpreting the 'from' point: The bearing 'of X from Y' means you must draw your North line and measure your angle starting at Y.
- ×Unit mismatch in scales: When finding a scale ratio like 1:n, you must convert both the map distance and real distance to the same unit (e.g., both to cm) before simplifying the ratio.
- ×Incorrectly calculating back-bearings: A common error is subtracting the bearing from 360°. Remember that North lines are parallel, so you must add or subtract 180°.
- ×Visual misjudgement: Relying on the appearance of your sketch instead of calculating angles using geometric rules. Your sketch is a guide, not a precise drawing.
No-calculator tips
- ✓Always draw a large, clear diagram. Mark all North lines, known angles, and distances. This is the most crucial step for solving these problems.
- ✓Look for simple geometric shapes. Questions are often designed to create right-angled triangles, isosceles triangles, or equilateral triangles.
- ✓Remember key trigonometric values for special angles like 30°, 45°, and 60°. For example, sin(30)=0.5, cos(60)=0.5, tan(45)=1. Problems are often set up to use these values.