Pythagoras and Trigometry

Higher Level Students: Please go down further to your section.

Foundation Level. At the Foundation level there are no questions on Trigonometry (sines, cosines, tangents and all that stuff), but there is plenty to be careful about with Pythagoras’ Theorem.

Questions on Pythagoras are normally very easy to get if you know what you are doing because there are only a small number of ways these questions can be asked and if you have lots of practice, you are almost guaranteed to get full marks.

1. The first thing to remember is that Pythagoras’ Theorem can only be applied to right angled triangles, but sometimes you sense the question is about Pythagoras’ Theorem, but there is no right angle in sight (or at least not where you want it to be). However, a right angled triangle can normally be found if you look carefully enough.

One common example is that you are given an isosceles triangle. This may be the front of a tent or a flag pole held up with cables, for example. The thing to do here is to draw a line down the middle of the isosceles triangle and, bingo, you have two identical right angled triangles!

In this tent front, for example, drawing a line down the middle gives two right angled triangles 1.5 metres wide and 2.5 metres high. Now you can find the length of the hypotenuse or the area of each triangle.

Another typical variation is to be given a small building such as a shed or lean-to. Here, drawing a line parallel to the floor and passing through the lowest point of the roof gives a right angled triangle.

This construction line gives a right angled triangle 0.7 metres high and 5 metres wide (same as the floor).

Now you can calculate:

a) The sloping length of the roof using Pythagoras’ Theorem √(0.7² + 5²) =  √25.49 = 5.048 = 5.05 metres (3 Sig Figs)

b) The area of the triangle ½ x 0.7 x 5 = 1.75 square metres

c) The area of the rectangle 1.9 x 5 = 9.5 square metres

d) The total area of the side of the building 1.75 + 9.5 square metres.

How easy is that?!

Sometimes you have an isosceles triangle on top of a rectangle as in the frontage of a typical garage:

I am sure you can see now how to divide this with a single horizontal line into an isosceles triangle similar to the tent shape and a rectangle. Then you can divide the triangle into two right angled triangles by dropping a vertical line from the top (apex) of the garage roof.

Then you can calculate anything you like: the length of the roof, the area of the triangles and the area of the rectangle. From the last two you can calculate the area of the whole of the front of the garage.

Other examples include questions involving bearings, sightings from the tops of cliffs etc, but there must be right angled triangles in there somewhere which you can find by just adding a few construction lines.

If not, it’s not a Pythagoras question. But if you think about it, what other methods have you been taught to calculate the lengths of lines in these types of diagrams? There aren’t any others.

Need some questions to practise? Go to www.gcsemathematics4u.com

Higher level

Please make sure you are familiar with the ideas in the Foundation Level discussion above. Remember that the Higher Level syllabus includes all of the Foundation Level syllabus!

If you are taking the higher level, you have an additional problem that the Foundation Level students don’t have. You know two ways to solve problems involving right angled triangles, Pythagoras’ Theorem and trigonometry, so the first thing you need to decide is which one to use.

This is surprisingly easy (and pretty obvious once you have realised it). If the triangle involves angles and you are either asked to find the length of a side or the value of an angle, it’s normally trigonometry. If there are no angles mentioned and you are not required to find one it must be Pythagoras’.

But you need to be careful. Sometimes you are given angles as well as the lengths of sides because the questions have more than one part. In the first part you may be asked to use two sides to find a third (which is obviously Pythagoras) and in another part you may need to use the angle information or find an angle (which is trigonometry). So you need to think very carefully about which information you need for each part of the question to help you decide whether to use trigonometry or Pythagoras’ Theorem.

But that’s why it’s called Higher Level.

You, too, may have all sorts of questions where you need to draw some construction lines as with the Foundation Level questions, plus some of your questions may need the answer from one part to calculate the next part or may use the same idea twice in one question. Bearing questions are a good example of this, where a ship, for example, makes the first part of its journey on one bearing and then changes direction for the second part:

Here we have a boat sailing 25 Km on a bearing of 124⁰ and then 32 Km on a bearing of 38^o.    Normally we are asked to find the total distance travelled east and the total distance travelled north etc. A few construction lines will soon give you all the right angled triangles you need:

As with the Foundation level, these are easy marks to get because the number of variations is limited and if you practise them enough you will sail thorough them (sorry about the pun!).

Need some questions to practise? Go to www.gcsemathematics4u.com