Pythagorean Theorem

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Pythagorean Theorem

Pythagorean theorem is used in a right angle triangle to calculate the sides of the triangle. When we are given two sides length in a right angle triangle we can find the missing side by using the Pythagorean theorem. Also to prove if a triangle is a right angle triangle.

What does Pythagorean Theorem says?

In a right angle triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.

$\displaystyle {{a}^{\text{2}}}+{{b}^{\text{2}}}~={{c}^{\text{2}}}$

a and b represents the two sides of the triangle.
c represents the longest side the hypotenuse, the side opposite the right angle.

pythagorean

The Pythagorean theorem can also be formulated as “The area of the square” whose side is the hypotenuse is equal to the sum of the areas of the squares of the other two sides.

pythagorean theorem

Proof of the Pythagorean Theorem

There are some proofs of this famous theorem but we are going to explain the one using Algebra

How can this be proved using Algebra?

This theorem can be proved algebraically using four copies of a right angle triangle with sides a, b and c, arranged symmetrically around a square with side c.

pythagorean theorem

The larger square that is formed has the side: a + b.
The big square area is: $\displaystyle A={{\left( a+b \right)}^{2}}$.
The area of each triangle is: $ \displaystyle A=\frac{a\cdot b}{2}$.
The area of the small square: $ \displaystyle A=c\cdot c={{c}^{2}}$

The four triangles and the square inside with the side c must have the same area as the big square.

Big square = small square + four triangles

$ \displaystyle {{\left( a+b \right)}^{2}}={{c}^{2}}+\frac{ab}{2}+\frac{ab}{2}+\frac{ab}{2}+\frac{ab}{2}$

$ \displaystyle {{\left( a+b \right)}^{2}}={{c}^{2}}+\frac{4ab}{2}$

$ \displaystyle {{\left( a+b \right)}^{2}}={{c}^{2}}+2ab$

From which:
$ \displaystyle {{c}^{2}}={{\left( a+b \right)}^{2}}-2ab$

Now expand the known formula: $ \displaystyle {{\left( a+b \right)}^{2}}$
And see if we can get the Pythagoras theorem:
$ \displaystyle {{c}^{2}}={{a}^{2}}+{{b}^{2}}+2ab-2ab$

$ \displaystyle {{c}^{2}}={{a}^{2}}+{{b}^{2}}$

This is the most easy proof of the Pythagorean theorem : $ \displaystyle {{c}^{2}}={{a}^{2}}+{{b}^{2}}$

Example 1: Find the missing side in the right angle triangles

pythagorean theorem

Solution:
1. At the first triangle we see that the missing side is the hypotenuse c.
Using directly the Pythagorean theorem:

pythagorean theorem

$ \displaystyle {{c}^{2}}={{a}^{2}}+{{b}^{2}}$
$ \displaystyle {{c}^{2}}={{\left( 7 \right)}^{2}}+{{\left( 8 \right)}^{2}}$
$ \displaystyle {{c}^{2}}=49+64$
$ \displaystyle {{c}^{2}}=113$

$ \displaystyle c=\sqrt{113}$

So, we found the missing side by using the Pythagorean theorem

2. At the second triangle we see that the missing side is one of the sides b.
We can’t use the Pythagorean Formula Directly but we are going to transform it so we can isolate the side which we need to find.

$ \displaystyle {{b}^{2}}={{c}^{2}}-{{a}^{2}}$
$ \displaystyle {{b}^{2}}={{\left( 10 \right)}^{2}}-{{\left( 6 \right)}^{2}}$
$ \displaystyle {{b}^{2}}=100-36$
$\displaystyle {{b}^{2}}=64$
$ \displaystyle b=\sqrt{64}$
b = 8

So, we found the side by using the Pythagorean theorem.

We can also use the theorem to determine if a triangle is right-angled or not. Substitute the values of a, b and c of the triangle into the formula and check to see if it fits. If + does not equal  then it is not a right angle triangle.

Example 2: Use Pythagoras’ theorem to decide whether or not the triangle shown below is right-angled.
Check to see if Pythagoras’ theorem is satisfied:

c² = a² + b²

13² = 10² + 6²

169 ≠ 100+36

The Pythagoras theorem is not satisfied, so the triangle is not a right angle triangle.

Example 3: Find the distance between the points A (3,6) and B(-4,8).

Solution: The difference between y coordinates:  AB = 8 – 6 = 2 units.

The difference between x coordinates:  AC = 3 – (-4) = 7 units.

Apply Pythagoras theorem.
BC2 = 22 + 72
= 4 + 49
= 53.
So, BC = $ \displaystyle \sqrt{53}$
= 7.28 units.

Pythagoras theorem can be used to solve real life problems. It is used frequently in architecture, constructing working or navigation.

Be careful!
You generally won’t be told to use Pythagoras theorem to solve problems. Always check for right-angled triangles in the context of the problem to see if you can you the theorem to solve it. It is usually useful to draw the triangle that you are going to use as part of your working.

Problem 1: The diagram shows a bookcase that has fallen against a wall. If the bookcase is 2 m tall,and it now touches the wall at a point 1.8 m above the ground,calculate the distance of the foot of the bookstore from the wall. Give your answer to 2 decimal places.

Solution: Think what triangle the situation would make and then draw it. Label each side and substitute the correct sides into the formula.

Apply the Pythagoras’ theorem:

a2 + b2 = c2

x2 + 1.82 = 22

x2 = 2– 1.82

= 4 – 3.24

= 0.76

x = $ \displaystyle \sqrt{76}=0.87m$

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