Mr Fs Maths NotesTrigonometry

1. Pythagoras

1. PythagorasWhat is Pythagoras Theorem?

Pythagoras Theorem is probably the most famous theorem in the history of mathematics It was invented by a Greek named Pythagoras (or one of his loyal followers who always marked any of their discoveries with the Pythagoras brand) somewhere around 6BC Pythagoras discovered a very important relationship between the lengths of sides in a right-angled triangles:

If you take the lengths of the two shortest sides of any right-angled triangle, square them and add the answers together, you end up with the square of the longest side (the hypotenuse)

2. What is the Hypotenuse?

In order to use Pythagoras Theorem (or all the trig that is coming around the corner!), you must be an expert at finding the Hypotenuse of any right angled triangle

The Hypotenuse is the longest side of the right-angled triangle, and it is the side opposite the right-angle!3. The two forms of Pythagoras Theorem

Pythagoras Theorem can be written in two ways depending on whether you want to find the length of the hypotenuse of a triangle, or one of the other sides.

The two ways are just different arrangements of the same original formula, so if you are good at formula re-arranging, then you only need to remember one!hypotenusehypotenusehypotenuse

4. Finding the Hypotenuse

1. Label the Hypotenuse h, and the other sides a and b

2. Use the following formulae:

h2 = a2 + b2

3. Replace the letters with the numbers you have been given, and carefully do the sum!hab

5. Finding a side that isnt the Hypotenuse

1. Label the Hypotenuse c, label the side you want to find a, and the other side b

2. Use the following formulae:

a2 = h2 - b2

3. Replace the letters with the numbers you have been given, and carefully do the sum!

Note: As I mentioned before, this version of the formula is just a different arrangement of:h2 = a2 + b2

Just subtract b2 from both sides and you should see what I mean!hab

Examples9 cm11 cm?Okay, so the side we want to find is the Hypotenuse, so lets go through our routine:1. Label the sides2. Use the formula:3. Put in the numbers:cabNote: Our answer is longer than both our other sides which is good because the hypotenuse is supposed to be the longest side!1.Square root both sides!

3.1 m10.2 m?Okay, so the side we want to find is NOT the Hypotenuse, so lets go through our routine:1. Label the sides2. Use the formula:3. Put in the numbers:cabNote: Our answer is shorter than our hypotenuse which is good because the hypotenuse is supposed to be the longest side!2.Square root both sides!

Its just a right angled triangle and we want to find a side that is NOT the Hypotenuse, so lets go through our routine:1. Label the sides2. Use the formula:3. Put in the numbers:3.Square root both sides!A 5m ladder rests against the side of a house. The foot of the ladder is 1.5m away from the house. How far up the side of the house does the ladder reach?At first glance this question does not appear to have anything to do with Pythagoras, but in these sort of situations, always follow this advice: IF ITS TRICKY, DRAW A PICCY!!! and then look what we have!5 m1.5 m ?acb

Its just a right angled triangle and we want to find the Hypotenuse, so lets go through our routine:1. Label the sides2. Use the formula:3. Put in the numbers:4.Square root both sides!Find the distance between these two co-ordinates: (4, 5) and (-2, 1)Again, at first glance this question does not appear to have anything to do with Pythagoras, but if we do a quick sketch of our co-ordinates, then look what we have!(4, 5)(-2, 1)cabTo work out the lengths of the sides, we just count how many squares would be in between on a co-ordinate grid!47

Good luck with your revision!

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