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4.2 E) Construction & Loci Problems
4.2 E) Construction & Loci Problems
We are going to have a look at a question that involves constructing perpendicular lines and loci. Make sure that you have covered all of the content in the previous sections in this whole section before working through this section.
Click here for a printable version of this question.
Click here for a printable version of this question.
Example 1
The diagram below shows an office. The scale of the diagram is 1 cm = 3 metres.
The diagram below shows an office. The scale of the diagram is 1 cm = 3 metres.
The office has purchased a new printer and they are deciding where to put the printer in the office. They want to make sure that the location of the printer satisfies the following:
Find the area on the above diagram where the printer can be located.
We answer question like this by drawing each of the conditions seperately. We then find the area on the diagram where both of the bullet points are satisfies.
Let’s start by working through the first condition, which is that the printer needs to be closer to B than A. We are able to find where the printer is closer to B than A by drawing a perpendicular bisector between A and B. This is because all of the points on a perpendicular bisector will be equidistance from point A and B. Therefore, after we have the perpendicular bisector, we can find the area on the diagram that is closer to B than A.
We draw a perpendicular bisector between two points by opening a compass out so that the distance between the point and pencil on the compass is greater than half the distance between point A and B. The next step is to place the point of the compass on point A and draw a construction line. We then do the same for point B. The construction lines are shown on the diagram below.
- The printer needs to be closer to B than A.
- The printer needs to be within 12 m of D.
Find the area on the above diagram where the printer can be located.
We answer question like this by drawing each of the conditions seperately. We then find the area on the diagram where both of the bullet points are satisfies.
Let’s start by working through the first condition, which is that the printer needs to be closer to B than A. We are able to find where the printer is closer to B than A by drawing a perpendicular bisector between A and B. This is because all of the points on a perpendicular bisector will be equidistance from point A and B. Therefore, after we have the perpendicular bisector, we can find the area on the diagram that is closer to B than A.
We draw a perpendicular bisector between two points by opening a compass out so that the distance between the point and pencil on the compass is greater than half the distance between point A and B. The next step is to place the point of the compass on point A and draw a construction line. We then do the same for point B. The construction lines are shown on the diagram below.
The final step for drawing the perpendicular bisector is to draw a line that passes through both of the points where the construction lines intercept.
The question says that the printer needs to be closer to B than A. The area where this is true is below the perpendicular bisector that we have just drawn; I have shaded this area blue.
We are now onto the second condition in the question, which is that the printer must be less than 12 metres away from D. We are able to find the area that is less than 12 metres from D by creating a circular locus from D. Before we are able to draw the locus from D, we need to find out what 12 metres is on the diagram. We find out what 12 metres is on the diagram by dividing 12 by the scale. The scale for this question is 1 cm = 3 m. Therefore, we undertake the following calculation:
This tells us that the printer on the diagram must be within 4 cm of D. We find the area that is less than 4 cm from D by opening a compass up so that the distance between the point and the pencil on the compass is 4 cm. We then place the point of the compass on point D and spin the compass around. The outcome is shown below.
We now have the areas where each of the two bullet points hold. However, we want to find the area whereby both of the above points hold. We do this by looking for the overlapping areas and this area is labelled on the diagram below.
Therefore, the printer should be placed in the labelled area.