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}

- 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:

- 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:

If (Hypotenuse)^{2} = (Shortest side) ^{2} + (Other side)^{2}

Then the triangle is right angled.

Example

Is this a right angled triangle ?

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:

A_{1} is a rectangle,

so

A_{2} is a triangle,

so

To find the perpendicular height, x

Use Pythagoras' Theorem

Substituting into the equation for A_{2} :

Thus the area of the painted surface

A_{1} + A_{2} = 50 + 16.6 = 66.6 m^{2}

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.