The diagram below shows the
positions of four points 𝐴, 𝐵, 𝐶, and 𝐷 on a map. 𝐴𝐵 is parallel to 𝐶𝐷. 𝐷𝐴 is parallel to 𝐵𝐶. The bearing of 𝐵 from 𝐴 is 082
degrees. Angle 𝐴𝐵𝐶 is 43 degrees. Work out the bearing of 𝐷 from
All bearings are measured clockwise
from north. This means that the bearing of 𝐵
from 𝐴 of 082 degrees can be drawn on the diagram as shown. As the quadrilateral is made up of
two pairs of parallel sides and the angles are not 90 degrees, we can say that
𝐴𝐵𝐶𝐷 is a parallelogram.
Adjacent angles in a parallelogram
add up to 180 degrees. Therefore, angle 𝐴𝐵𝐶 and angle
𝐵𝐴𝐷 equal 180 degrees. We know that angle 𝐴𝐵𝐶 is 43
degrees. Therefore, 43 plus angle 𝐵𝐴𝐷 is
equal to 180. Subtracting 43 from both sides of
this equation gives us 𝐵𝐴𝐷 is equal to 137. Therefore, the angle on the diagram
𝐵𝐴𝐷 is equal to 137 degrees.
The question asked us to work out
the bearing of 𝐷 from 𝐴. This is shown on the diagram by the
pink arrow. Again, we must go clockwise from
north. This angle is equal to 82 degrees
plus 137 degrees. 82 plus 137 is equal to 219. Therefore, the bearing of 𝐷 from
𝐴 is 219 degrees.
We can check our answer is sensible
by considering the four key points on a compass. If a point was due east, it would
have a bearing of 90 degrees. If it was due south, 180 degrees
and if it was due west, 270 degrees. On the map, it is clear that point
𝐷 is on a bearing from 𝐴 between south and west. Therefore, the answer must be
between 180 and 270 degrees. This means that our answer of 219
degrees is sensible.