Intersecting loci
Problems involving loci may require you to draw a number of constructions and identify a point or area that satisfies all conditions.
Example
The diagram shows a sketch of a garden that measures 6 m by 7 m.
A flowerbed is to be planted. It must be closer to the tap than the electricity. It must also be less than 3 m away from the tap so that it can be watered easily. Identify the area in which the flowerbed can be planted.
To ensure it is closer to the tap than electricity, we need to bisect the line going from the tap to the electricity.
We can discount the area that is closer to the electricity point. Then create the loci of points 3 cm away from the tap.
We can discount the area more than 3 cm away from the tap. This leaves the labelled area where the flowerbed can be located.
Question
A boat is preparing to anchor down. To rest safely, the boat must be closer to the beach than to the marshland, and should be at least 1 ½ miles away from the marshland. Identify the area the boat should stop. The sketch is given using the scale 1 cm = 1 mile.
Draw a parallel line that is 1.5 cm from the line EF. All points to the right of this line are at least 1 ½ miles away from the marshland.
Bisect the angle DEF. All points above this line are closer to the beach than to the marshland.
Shade the region of points that satisfies both the criteria to indicate where the boat should settle.
