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Constructions & Loci (AQA GCSE Maths)
Revision Note
Constructions
What are constructions?
- A construction is a process where you create particular geometric objects using only a pair of compasses and a straight edge
- It is an excuse to get the maths toys out – rulers and compasses in particular!
- You will often be working with scale drawings with this topic
1. Perpendicular bisector
- This is a line that cuts another one exactly in half (bisect) but also crosses it at a right angle (perpendicular)
- It shows a path that is equidistant (equal distance) between two places
- If you are on one side or the other of this line you know you are closer to one place than the other
- To construct a perpendicular bisector, you will need a sharp pencil, ruler and set of compasses
- You may be asked to construct a perpendicular bisector of a line segment or to find the region that is closer to one point than another, the following method is used for both
- STEP 1
Set the distance between the point of the compasses and the pencil to be more than half the length of the line
- STEP 2
Place the point of the compasses on one end of the line and sketch an arc above and below the line
- STEP 3
Keeping your compasses set to the same distance, move the point of the compasses to the other end of the line and sketch an arc above and below it again, the arcs should intersect each other both above and below the line
- STEP 4
Connect the points where the arcs intersect with a straight line
- Final result:
2. Perpendicular to a line, from a point
- This is drawing a line that is perpendicular to another one but the line has to go through a particular point
- It is essentially the same as the perpendicular bisector with one extra step to do first
- They show the shortest distance from the point to a line
- eg. the shortest distance from a lighthouse to the coastline
- To construct a perpendicular from a point to a line, you will need a sharp pencil, ruler and set of compasses
- You may be asked to construct a perpendicular from a point to a line or to find the shortest distance from a point to a line, the following method is used for both
-
- STEP 1
Set the distance between the point of your compasses and the pencil to be greater than the distance between the point P and the line
- STEP 2
Placing the point of the compasses onto the point P, draw an arc that intersects the line in two places
- STEP 3
Make sure that the distance between the point on the compasses and the pencil is greater than half the distance between the two points of intersection, place the point of the compasses on to one point of intersection and sketch an arc above and below the line
- STEP 4
Keeping the distance between the point on the compasses and the pencil the same, move th point of the compasses onto the other point of intersection and again draw an arc both above and below the line, these should intersect with the previous arcs
- STEP 5
Join the points where the arcs intersect each other and the original point P with a straight line
- STEP 1
- Final result:
3. Angle bisector
- This is a line that cuts an angle exactly in half (bisect),
- Either side of the line shows you are closer to one of the angle lines than the other
- It is essentially another variation on the perpendicular bisector
-
It shows a region on a map/diagram that is closer to one side than another
- To construct an angle bisector, you will need a sharp pencil, ruler and set of compasses
- You may be asked to construct a bisector of an angle or to find the region that is closer to one side of an angle than the other, the following method is used for both
- STEP 1
Open the compasses, the distance between the point and the pencil is not particularly important but setting them about half the distance of the lines forming the angle is reasonable
- STEP 2
Placing the point of the compasses at the point of the angle, sketch an arc that intersects both of the lines that form the angle
- STEP 3
Set your compasses to the distance between these two points of intersection, place the point of the compasses on one of the points of intersection and sketch an arc
- STEP 4
Keeping the distance between the point of the compasses and the pencil the same, place the point of the compasses on the other point of intersection and sketch an arc, this should intersect the arc sketched in STEP 3
- STEP 5
Join the point of the angle to the point of intersection with a straight line
- Final result:
Examiner Tip
- Make sure you have all the equipment you need for your maths exams, along with a spare pen and pencil
- An eraser and a pencil sharpener can be essential on these questions as they are all about accuracy
- Make sure you have compasses that aren’t loose and wobbly and make sure you can see and read the markings on your ruler and protractor
Worked example
On triangle ABC below, indicate the region that is closer to the side AC than the side BC.
This question is asking for the region that is closer to one side of an angle than the other, so an angle bisector is needed.
Open your set of compasses to a distance that is approximately half the length of the sides AB and AC.
This distance is not too important, but keeping it the same length throughout the question is very important.
Place the point of the compasses at A and draw arcs across the lines AB and AC. Be very careful not to change the length of the compasses as you draw the arcs.
Leaving the compasses open at the same length, put the point at each of the places where the arcs cross the sides AB and AC and draw new arcs which cross over each other in the middle.
Draw a line from A to the point where the arcs cross over each other (this will not usually be directly on the third side of the triangle and never has to be!)
Shade the region between the angle bisector (the line you have drawn) and the side AC.
Loci
What are Loci?
- A locus (loci is plural) is a line/shape/path that is determined by following a rule
- eg. always being 2 m away from a wall
- You may be asked to construct a locus, although the language used in exam questions won’t always mention these words as questions are often based on real-world situations
- You may be expected to use some of the constructions mentioned in the revision note on constructions
- You may need to use a ruler, a protractor or a pair of compasses
What are the common types of loci?
- Points that are a fixed distance from a point
- This will form a circle around the point
- Use a pair of compasses
- This will form a circle around the point
- Points that are a fixed distance from a line
- This will form a race track shape
- Two parallel lines
- Two semi-circles at the end of the lines
- This will form a race track shape
- Points that are equidistant from two points
- This will form the perpendicular bisector between the line segment between the two points
- Points that are equidistant from two lines
- This will form the angle bisector of the angle between the two lines
Worked example
A house lies between Town A and Town B as shown on the scale diagram below.
Two masts, located at the points R and S, provide the area shown on the map with radio signals.
The house will receive its radio signal for the mast located at point R if it is either...
... closer to Town A than Town B, or...
... outside a region 5 miles from the mast at point S.
Showing your working carefully on the scale diagram below, determine whether the house receives radio signals from the mast at point R or the mast at point S.
Begin by finding the region satisfying the first condition, that the house is closer to Town A than Town B.
This is found by constructing the perpendicular bisector of the line segment that joins Town A and Town B. You will need to add this line segment in yourself before starting.
Open your compasses to more than half of the distance from Town A to Town B and draw arcs both above and below the line joining Town A to Town B.
Do this from both the point at Town A and the point at Town B. The arcs should cross over each other.
The perpendicular bisector is the line that passes through both of the points where the arcs cross over each other.
The house is on the same side of the perpendicular bisector as Town B is.
The house is closer to Town B than Town A.
Now check the second condition, to see if the house is further than 5 miles from the mast at point S.
To find the locus of points exactly 5 miles from point S, first consider the scale given on the scale drawing.
1 cm = 1 mile
5 cm = 5 miles
Open your set of compasses to exactly 5 cm. Measure this carefully using a ruler.
Being extra careful not to change the length of your compasses, put the point at S and draw a circle around S with a radius of 5 cm. You may not be able to draw the full circle, but make sure you have the part that is near the point where the house is located.
The house is located outside of this region, so it is more than 5 miles from the mast at the point S.
The house only satisfies one of the conditions given to receive its signal from the mast at point S.
The house receives its radio signal from the mast at point R.
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