Constructing Triangles

What are triangle constructions?

In mathematics a construction is an accurate drawing that normally uses equipment such as
- a sharp pencil
- a ruler
- a protractor and/or a pair of compasses
You will be given information about the size of some of the angles or the lengths of some of the sides of the triangle you are being asked to draw
Depending on the type of triangle you are being asked you draw you will need to follow a specific method and you may need different equipment
The types of triangle you may be asked to construct include
- SSS – you are given the lengths of all three sides
- SAS – you are given the lengths of two sides and the angle in between them (the included angle)
- ASA – you are given the size of two angles and the length of the side in between them (the included side)

How do I construct an SSS triangle?

If you are given all three sides of a triangle, you will need a pencil, a ruler and a pair of compasses
STEP 1
Use a ruler to draw the longest side as the horizontal base near the bottom of the space you have been given
- This needs to be accurate, measure it carefully with your ruler
- Write its length (with units) just underneath

STEP 2
- Using your ruler to measure, open your compasses so that the length from the compass point to the tip of your pencil is exactly the length of one of the remaining sides
- Being extra careful not to change the length, put the compass point on one end of the horizontal line you have drawn and draw an arc above the horizontal line
STEP 3
- Using your ruler to measure again, open your compasses so that the length from the compass point to the tip of your pencil is exactly the length of the third side
- Being extra careful not to change the length, put the compass point on the other end of the horizontal line and draw another arc, making sure that it crosses the first arc
STEP 4Use your ruler to draw straight lines from each end of the horizontal line to the point where the arcs cross over
STEP 5
Use your ruler to check that the two new lines are exactly equal to the lengths given in the question
- When you are confident that they are accurate, label the lines

It is important that you do not rub out your arcs as the examiner will use these to check your work
Sometimes the instructions will include a triangle name such as triangle ABC
- Make sure you label each vertex with the correct letters

How do I construct an SAS triangle?

If you are given two sides of a triangle and the angle in between them, you will need a pencil, a ruler and a protractor
STEP 1
Use a ruler to draw the longest given side as the horizontal base near the bottom of the space you have been given
- This needs to be accurate, measure it carefully with your ruler
- Write its length (with units) just underneath

STEP 2
- Place the centre point of the protractor on one end of the side that you have just drawn, measure the given angle from the side and make a mark to indicate where it is
- Draw a straight line from where you had placed the protractor through the mark and extend it further

STEP 3
- Measure along the line you have just drawn in STEP 2 from the end at which it connects to the first horizontal line
- Make a mark on the line when you have measured the length of the second given side

STEP 4Use your ruler to draw a straight line from other end of the first horizontal line to the mark you have just made on the second line
STEP 5
Use your protractor and ruler to check that the measured angle and sides are exactly equal to the sizes given in the question
- When you are confident that they are accurate, label the sides and the angle

It is important that you do not rub out your construction lines as the examiner will use these to check your work
Sometimes the instructions will include a triangle name such as triangle ABC
- Make sure you label each vertex with the correct letters

How do I construct an ASA triangle?

If you are given two angles of a triangle and the length of the side in between them, you will need a pencil, a ruler and a protractor
STEP 1
Use a ruler to draw one of the sides as the horizontal base near the bottom of the space you have been given
- This needs to be accurate, measure it carefully with your ruler
- Write its length (with units) just underneath
STEP 2
- Place the centre point of the protractor on one end of the side that you have just drawn, measure the given angle from the side and make a mark to indicate where it is
- Draw a straight line from where you had placed the protractor through the mark and extend it further

STEP 3
- Place the centre point of the protractor on the other end of the first side that was drawn and measure the given angle indicating its position with a mark
- Draw a straight line from where you had placed the protractor through the mark and extend it further
- This line should cross the line you drew in STEP 2, if it doesn’t, extend your lines further

STEP 4Use your ruler to draw straight lines from each end of the first horizontal line to the point where the lines drawn in STEP 2 and STEP 3 cross over
STEP 5
Use your protractor to check that the two measured angles are exactly equal to the sizes given in the question
- When you are confident that they are accurate, label the angles

It is important that you do not rub out your construction lines as the examiner will use these to check your work
Sometimes the instructions will include a triangle name such as triangle ABC
- Make sure you label each vertex with the correct letters

Exam Tip

To ensure you get full marks in your constructions questions
- Make sure you are confident using your compasses
- Make sure that your compasses are not loose
- Do not erase the construction arcs from your diagram

Worked example

Using a ruler and pair of compasses only, construct a triangle with sides 6 cm, 7 cm and 10 cm.
Leave in your construction arcs.

Draw the 10 cm line as the horizontal base.
Place the point of the compasses at each end and draw an arc with radius 6 cm from one end and another with radius 7 cm from the other end.
The third vertex of the triangle is the point at which they intersect.

Use your ruler to measure each side and check for accuracy.

Worked example

Using a ruler and a protractor only, construct a triangle with sides 9 cm and 6 cm and an included angle of 62 ^o.

Draw the 9 cm line as the horizontal base.

Place the centre of the protractor at one end of the horizontal line and measure 62 ^o.

Measure 6 cm along the new line and indicate it with a mark.

Use your ruler to draw a straight line connecting the other end of the horizontal line to the mark.
Label the lengths of the sides and the angle that you are given in the question.

Worked example

Using a ruler and a protractor only, construct a triangle with angles 36 ^o and 59 ^o and an included side of length 8 cm.

Draw the 8 cm line as the horizontal base.

Place the centre of the protractor at one end of the horizontal line and measure 36 ^o.

Put the centre of the protractor over the other end of the horizontal line and measure 59 ^o.

Using your ruler, join each end of the horizontal line to the point where the other two lines intersect.
Label the size of the angles and the length of the side that you were given in the question.

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