Lesson 9: Areas of 2-D shapes (I)May 7, 2011
1. Pythagorean Theorem
A right-angled triangle is a triangle that has a right angle. We name hypotenuse to the side opposite the right angle (c on the picture).
In a right-angled triangle the sum of the squares of the lengths of the sides equals the square of the length of the hypotenuse.
- If we know the lengths of a side and hypotenuse, we can find the length of the other side
- And if we know the lengths of the two sides we can find the length of the hypotenuse
When the side lengths of right triangle satisfy the Pythagorean Theorem, these three numbers are known as “Pythagorean triples or triplets”
How to identify and create Pythagorean triples?
The most common examples of Pythagorean triples are:
- 3,4,5 triangles
- A 3,4,5 triple simply stands for a triangle that has a side of length 3, a side of length 4 and a side of length 5.
- If a triangle has these side length then it must be a right triangle
- 5,12,13 right triangles
- 7,24,25 right triangles
- 8,15,17 right triangles
We can conclude the following:
- Only right triangles verify the Pythagorean Theorem
- If a triangle has a triple as side lengths then it must be a right triangle
Any multiple of the ratios above represent the sides of a right triangle.
Here you are two interesting pages:
On this page you can check that:
- if you have that if you have an right-angled triangle:
a2 + b2 = c2
- if you have that if you have an acute-angled triangle:
a2 + b2 < c2
- if you have that if you have an obtuse-angled triangle:
a2 + b2 > c2
If you click on the picture below you will find an animation the geometric demonstration of the Pythagorean Theorem (go down the page because the animation is at the end)
Finally this is a video on Pythagorean Teorem and its Converse (Is this triangle a right triangle?)
2. Applications of the Pythagorean Theorem
There are many applications of this famous theorem. One of them are the following:
- Identifying right triangles
- Calculating the diagonal of a rectangle
- Calculating the height of an isosceles triangle
- Calculating the apothem of a regular polygon
If we remember the first I recommended, we can write the conditions that led us to classify the triangles according their angles:
- a2 + b2 = c2 , then it is a right-angled triangle
- a2 + b2 < c2 , then it is an acute-angled triangle
- a2 + b2 > c2 , then it is an obtuse-angled triangle
On the following links you can practice exercises on Pythagorean Theorem.
This is a video where is solved a problem by using this theorem