# Pythagoras’ Theorem

The longest side of a right- angled triangle is called the hypotenuse, which  is  always opposite the right-angle.

In any right- angled triangle,the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other sides.

For any right-angled triangle, this rule can be used to calculate the length of the hypotenuse if the lengths of the smaller sides are known.

(Hypotenuse)2 = (Shortest side)2  + (Other side)2

so
(Longest side)2 = (Shortest side)2  + (Other side) 2

## To find the length of the hypotenuse

• Sketch triangle
• Mark hypotenuse
• Write out pythagoras' theorem for the triangle

(Hypotenuse)2 = (Shortest side) 2  + (Other side) 2

• Solve
• Write out solution

Example

Find the length of the hypotenuse:

## To find the length of a shorter side

• Sketch triangle
• Mark hypotenuse
• Write out pythagoras' theorem for the triangle
(Hypotenuse)2 = (Shortest side)2  + (Other side)2
• Solve
• Write out solution

Example

Find the length of the missing side:

## The converse of Pythagoras

If  (Hypotenuse)2 = (Shortest side) 2  + (Other side)2
Then the triangle is right angled.

Example

Is this a right angled triangle ?

## Hidden Pythagoras

Very often, you will need to solve a question where the use of Pythagoras' Theorem does not seem obvious.

Example

Calculate the perimeter of triangle ABD. (Give your answer correct to 1 dp.)

Finding the perimeter requires the length of CD to be known.

Since ACB is a right angled triangle, Pythagoras' Theorem can be used to find length BC. Triangle BCD is also right angled, so Pythagoras' Theorem can be used again , with the value calculated for BC and the given 11 cm to find CD.

Finally, the lengths can be added to find the perimeter.

so

Thus

Perimeter = 12 + 11 +9 +7.6 = 39.6 cm(1 dp)

Example

The gable of a symmetrical building is painted yellow.

Calculate the area of the painted surface.

This is a composite area, so split into two parts:

A1 is a rectangle,

so

A2 is a triangle,

so

To find the perpendicular height, x

Use Pythagoras' Theorem

Substituting into the equation for A2 :

Thus the area of the painted surface

A1 + A2 = 50 + 16.6 = 66.6 m2

## Pythagoras with co-ordinates

Example

Calculate the length of the line that joins the points

A(-5, 10) and B ( 3 ,0 )

Solution :

Plot the points and draw the line.

Complete the right angled triangle

solve using Pythagoras' Theorem

This is the basis for the distance formula , which is part of Higher Mathematics Applications.