Unit 3 Section 5 : Constructions and Angles

This section looks at the constructing triangles when you know the lengths of the three sides.
It also introduces a way of classifying triangles according to the types of angles in them.

Constructing triangles
If we know the lengths of the three sides of a triangle, we can construct it on paper using a ruler and a pair of compasses.
The slideshow below shows how to do this construction. Use the left and right arrow buttons to move through the slideshow.

Constructing a triangle when the lengths of 3 sides are known
Use the arrows on either side to move through the slideshow

We could have worked out that this triangle was 'right-angled' by testing it with Pythagoras' theorem:
Squaring the two shorter sides and adding them together gives 225 and squaring the longest
side also gives 225, so Pythagoras' theorem is true for this triangle - it must be right-angled.

What about other triangles? What can we find out about other types of triangle using Pythagoras' theorem?

Types of angles in triangles
When considering the angles in triangles, there are three types of triangle: right-angled, obtuse-angled and acute-angled.

A right-angled triangle has one 90° angle (the other two angles are acute).
An obtuse-angled triangle has one obtuse angle (the other two angles are acute).
An acute-angled triangle has three acute angles.

The examples below show one of each type of triangle.
In each case the two shorter sides are marked a and b and the longest side is marked c.

Using the lengths of the sides to work out the type of triangle
From the diagram above, we can use the following method to calculate the type of triangle:

  1. Make c the length of the longest side, and calculate c².
  2. Make a and b the length of the two shorter sides, and calculate a² + b².
  3. Compare c² and a² + b².
  4. Use the table below to work out the type of triangle:
    c² = a² + b²right-angled triangle
    c² > a² + b²obtuse-angled triangle
    c² < a² + b²acute-angled triangle

Example Question
Use the method above to decide whether the triangles below are right-angled, obtuse-angled or acute-angled.
Once you have worked it out, click the Click on this button below to see the correct answer button to see whether you are correct.
(a) Triangle with sides 6cm, 7cm, and 8cm
(b) Triangle with sides 5cm, 12cm, and 13cm
(c) Triangle with sides 6cm, 11cm, and 14cm

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

Question 1
For this question, you will need some paper and equipment for constructing triangles.
Each of the sketches in the question shows a triangle with the lengths of sides marked.

Construct the triangles on paper and measure the angles A, B and C.
Measure the angles as accurately as possible; the computer will allow for a small inaccuracy but not much!
(a) A = °
B = °
C = °
(b) A = °
B = °
C = °
(c) A = °
B = °
C = °

Question 2
Decide if the triangles below are right-angled, obtuse-angled or acute-angled.
(a) Triangle with sides 10cm, 11cm and 14cm.
(b) Triangle with sides 10cm, 12cm and 16cm.
(c) Triangle with sides 9cm, 12cm and 15cm.
(d) Triangle with sides 8cm, 8cm and 8cm.
(e) Triangle with sides 2cm, 9cm and 9cm.
(f) Triangle with sides 9cm, 5cm and 5cm.
Question 3
Ahmed draws a square with sides of length 6cm.
He then measures a diagonal as 8.2cm.
 

Use Pythagoras' theorem to decide if he has drawn the square accurately.
Has he drawn it accurately?

Question 4
An isosceles triangle has a base of length 9cm and two equal sides of length 8cm.
 

Decide whether the angle θ is a right-angle, an acute angle, or an obtuse angle.

You have now completed Unit 3 Section 5
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Produced by A.J.Reynolds February 2003
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