Worked Examples in Science and Mathematics

Applying only to a triangle with one internal angle of 900 (a right angled triangle) the following definitions will apply (from the viewpoint of angle ‘a’):

A typical ‘right angled triangle’, or ‘right triangle’. The angles ‘a’ and ‘c’ are complementary, as the angle ‘b’ is 900 then angles ‘a’ and ‘c’ also add up to 900 since the internal angles of any triangle add up to 1800.

Before we look at two basic, simple triangles and the relationships between them, we perhaps ought to discuss Pythagoras Theorem:

“The square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides”

OK so what does this mean?

Take another look at our right angled triangle:

The green triangle has sides denoted A, B and C. Although they should actually be touching the sides, each square (representing the ‘square on that side’) has a side length equal to the corresponding triangle side length, and consequently an area of ‘that side length squared’.

Pythagoras Theorem states basically that in any right angled triangle the area of the largest square (on the hypotenuse) is equal to the sum of the areas of the remaining two side-squares.

Example:

Let us say that A = 4, B = 3 and C = 5, this is in fact one of the simplest right angled triangles and is not unsurprisingly referred to as a ‘3-4-5 triangle’

Given any two sides of a right angled triangle we can, by this theorem, evaluate the third with ease:

In this type of question you will be asked to “Find x”

Although it is quite funny, this answer probably won’t get you many marks in an exam:

Of course the correct answer is to square 7, square 4, add them together and find the square root of the sum.

Now you try:

Given this triangle:

It may be though, that you are not asked to find the hypotenuse, so squaring the two sides that you are given and adding them up etc. won’t be the right way.

If you are given the hypotenuse and another side you need to do this:

It doesn’t matter if the side ‘asked for’ is the shorter or longer of the two sides, as long as the hypotenuse is given the answer will be worked out the same way.

Over to you: