In these lessons, we will learn
 the Pythagorean Theorem
 the Converse of the Pythagorean Theorem
 how to use the Pythagorean Theorem
 proofs of the Pythagorean Theorem
Related topics:
More Geometry Lessons
What is the Pythagorean Theorem?
A right triangle consists of two sides called the
legs and one side called the
hypotenuse. The hypotenuse is the longest side and is opposite the right angle.
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The Pythagorean Theorem or Pythagoras' Theorem is a formula relating the lengths
of the three sides of a right
triangle.
If we take the length of the hypotenuse
to be c and the length of the legs
to be a and b then this theorem tells us that:
c^{2 }=^{ }a^{2}
+ b^{2}
Pythagorean Theorem states that
In any right triangle, the sum of the squared lengths of the two legs is equal to the squared length of the hypotenuse.
Note: Pythagorean theorem only works for right triangles
The following video gives the definition of Right Triangle and Pythagorean Theorem
What is the Converse of the Pythagorean Theorem?
The converse of the Pythagorean Theorem is also true.
For any triangle with sides a, b, c, if a^{2} + b^{2} = c^{2}, then the angle between a and b measures 90° and the triangle is a right triangle.
How to use the Converse of the Pythagorean Theorem?
We can use the converse of the Pythagorean Theorem to check whether a given triangle is an acute triangle, a right triangle or an obtuse triangle.
For a triangle with sides
a, b and
c and
c is the longest side then:
If
c^{2} <
a^{2} +
b^{2} then it is an acuteangled triangle, i.e. the angle facing side
c is an acute angle.
If
c^{2} =
a^{2} +
b^{2} then it is a rightangled triangle, i.e. the angle facing side
c is a right angle.
If
c^{2}>
a^{2} +
b^{2} then it is an obtuseangled triangle, i.e. the angle facing side
c is an obtuse angle.
Pythagorean Theorem Worksheets
The Pythagorean Theorem and The Converse of the Pythagorean Theorem
This video shows how to use the Pythagorean Theorem and its Converse to determine if a triangle is acute, right, or obtuse.
How to use the Pythagorean Theorem?
The Pythagorean Theorem can be used when we know the length of two sides of a right triangle and we need to get the length of the third side.
Example
1: Find the length of the hypotenuse of a right triangle
if the lengths of the other two sides are 3 inches and 4 inches.
Solution: 

Step
1: Write down the formula 
c^{2 }=^{
}a^{2} + b^{2} 
Step
2: Plug in the values 
c^{2}
= 3^{2} + 4^{2} 

c^{2}
= 9 + 16 

c^{2}
= 25 

c =


c = 5 
Answer:
The length of the hypotenuse is 5 inches.
Example
2: Find the length of one side of a right triangle if the
length of the hypotenuse is 10 inches and the length of the other
side is 9 inches.
Solution: 

Step
1: Write down the formula 
c^{2 }=^{
}a^{2} + b^{2} 
Step
2: Plug in the values 
10^{2} =
9^{2} + b^{2} 

100 = 81 + b^{2} 
Step
3: Subtract 81 from both sides 
19 = b^{2} 

b =


b ≈
4.36 
Answer:
The length of the side is 4.36 inches.
Have a look at the following video for more examples on how to use the Pythagorean theorem.
Proofs of the Pythagorean Theorem
There are many ways to proof the Pythagorean Theorem. We will look at three of them here.
How to proof the Pythagorean Theorem using Similar Triangles?
This proof is based on the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.
Given Triangle ABC drawn above in the image and prove a
^{2} + b
^{2} = c
^{2} using Similar Triangles
Triangle ABC ∼ Triangle ACD AA (Similarity Postulate)
Triangle ABC ∼ Triangle CBD AA (Similarity Postulate)
c/a = a/x (Converse of SSS Similarity Postulate)
c/b = b/y (Converse of SSS Similarity Postulate)
a
^{2} = cx (Cross multiply)
b
^{2} = cy (Cross multiply)
a
^{2} + b
^{2} = cx + cy (Adding the equations)
a
^{2} + b
^{2} = c(x + y) (GCF)
a
^{2} + b
^{2} = c
^{2} (Substitution)
How to proof the Pythagorean Theorem using Algebra?
In this proof, we use four copies of the right triangle, rearrange them and use algebra to proof the theorem.
How to proof of the Pythagorean Theorem using Rearrangement of shapes?
The following video shows how a square with area
c^{}^{}^{2} can be cut up and rearranged such that it can fit into two other smaller squares with areas
a^{2} and
b^{2}.
You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.
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