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

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In any right- angled triangle,the area of the square on the hypotenuseis 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 moment-g.comnverse 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 moment-g.comrrect 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 moment-g.commposite 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 moment-g.com-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.

See more: Us Liquid Gallons Per Hour To Liters Per Minute Conversion, Gallons (Us Fluid) Per Hour To Liters Per Minute moment-g.commplete the right angled triangle solve using Pythagoras" Theorem This is the basis for the distance formula , which is part of Higher Mathematics Applications.

### Pythagorean Triples : Listing

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