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
How do I construct a 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:
How do I construct an 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
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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.
