Converse of the Pythagorean Theorem
The converse of the Pythagorean Theorem helps you to find out if a triangle is a right triangle.
Basically, the converse of the Pythagorean Theorem states that whenever the sum of the squares of two sides equal to the square of the third side of the triangle, the triangle is a right triangle.
A few examples showing how to use the converse of the Pythagorean Theorem in order to determine is a triangle is a right triangle.
For example, given the following 3 sides, is the triangle a right triangle?
4, 5, 3
You just need to ask yourself the following question:
Is 5
^{2} = 4
^{2} + 3
^{2} ?
5
^{2} = 25 and 4
^{2} + 3
^{2} = 16 + 9 = 25
Since 25 = 25, the triangle with sides 4, 5, and 3 is a right triangle.
Notice that c must be equal to 5 since c is always the biggest side. After that it does not really matter which number is b or c. You can choose either 4 or 3 to be b or a.
2)
Do the sides 6, 8, and 10 form a right triangle?
You just need to ask yourself the following question:
Is 10
^{2} = 6
^{2} + 8
^{2} ?
10
^{2} = 100 and 6
^{2} + 8
^{2} = 36 + 64 = 100
Since 100 = 100, the triangle is a right triangle.
More examples showing how to use the converse of the Pythagorean Theorem:
3)
Do the sides 8, 12, and 14 form a right triangle?
You just need to ask yourself the following question:
Is 14
^{2} = 8
^{2} + 12
^{2} ?
14
^{2} = 196 and 8
^{2} + 12
^{2} = 64 + 144 = 208
Since 196 is not equal to 208, the triangle is not a right triangle.
4)
Do the sides 9, 12, and 15 form a right triangle?
You just need to ask yourself the following question:
Is 15
^{2} = 9
^{2} + 12
^{2} ?
15
^{2} = 255 and 12
^{2} + 9
^{2} = 144 + 81 = 225
Since 255 = 255, the triangle is a right triangle.
Here is a figure summarizing the converse of the Pythagorean theorem
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