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 solutionExample
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.
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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.