What is the Pythagoras Theorem?
The Pythagorean theorem states that, “In any right angled triangle, the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse”.
For a right angled triangle the legs are its shorter sides and the hypotenuse is its longest side. The shorter sides or the legs are also known as the perpendicular and the base of the triangle. The longest side, that is, the hypotenuse, is opposite to the right angle of the triangle.
Pythagorean Theorem Formula
As per the definition of the Pythagorean theorem we can use the following formula and apply it to any right angled triangle:
\(a^2~+~b^2~=~c^2\)
We will discuss this formula in detail in the next section.
How do we use the Pythagorean Theorem formula?
To apply the Pythagorean theorem formula you need the measure of at least two sides of a right angled triangle.
As we just learned that the square of the hypotenuse is the sum of squares of the legs.
Mathematically, the theorem can be expressed as:
\(a^2~+~b^2~=~c^2\)
Where,
- a and b are the lengths of the legs or the shorter sides
- c is the length of the hypotenuse or the longest side
The Pythagorean theorem formula can be used to find the measure of an unknown side in a right triangle. The theorem can also be applied to any triangle to determine whether it is right angled, or not.
Solved Examples
Example 1: Calculate the length of the hypotenuse of a right angled triangle whose lengths of shorter sides are 2 cm and 2.1 cm.
Solution:
Let a, b and c be the lengths of the sides of the right triangle, where a = 2 cm and b = 2.1 cm.
\(a^2~+~b^2~=~c^2\) [Write the Pythagorean theorem]
\((2)^2~+~(2 . 1)^2~=~c^2\) [Substitute 2 for a and 2.1 for b]
4 + 4.41 = \(c^2\) [Evaluate the power]
8.41 = \(c^2\) [Add]
2.9 = c [Take the positive square root of each side]
Hence, the length of the hypotenuse is 2.9 cm.
Example 2: Find the slant height of the triangular pyramid.
Solution:
\(a^2~+~b^2~=~c^2\) [Write the Pythagorean theorem]
\((8.8)^2~+~(6 . 6)^2~=~x^2\) [Substitute 8.8 for a, 6.6 for b and x for c]
77.44 + 43.56 =\(x^2\) [Evaluate the power]
121 = \(x^2\) [Add]
11 = x [Take the positive square root of each side]
Hence, the slant height of the pyramid is 11 ft.
Example 3: Find the measure of the unknown side length.
Solution:
The unknown side length is the leg b.
\(a^2~+~b^2~=~c^2\) [Write the Pythagorean theorem]
\((16)^2~+~b^2~=~(34)^2\) [Substitute 16 for a and 34 for c]
256 + \(b^2\) = 1156 [Evaluate the power]
\(b^2\) = 1156 – 256 [Subtract 256 from both sides]
\(b^2\) = 900
b = 30 [Take the positive square root of each side]
Hence, the unknown side length is 30 miles.
Example 4: The 3 sides of a triangle measure 26 units, 24 units and 10 units. Find whether this triangle is right angled or not.
Solution:
Here the longest side is 26 units.
24 units and 10 units are the legs or the shorter sides.
Let a be 24 units and b be 10 units.
\(a^2~+~b^2~=~c^2\) [Write the Pythagorean theorem]
\((24)^2~+~(10)^2~=~(26)^2\) [Substitute the values]
576 + 100 = 676 [Evaluate the power]
676 = 676 [Add]
So both sides are equal, that is, the given triangle satisfies the Pythagorean theorem.
Hence, the given triangle is a right angled triangle.
Example 5: The coordinate plane shows the location of three bakery shops in the market. Each unit on the grid plane represents 1 square mile. Determine the distance between shop A and shop C.
Solution:
Since each unit on the plane is 1 square mile, the side of this unit measures 1 mile. So as per the figure:
Side a = 3 miles
Side b = 4 miles
\(a^2~+~b^2~=~c^2\) [Write the Pythagorean theorem]
\((3)^2~+~(4)^2~=~c^2\) [Substitute 3 for a and 4 for b]
9 + 16 = \(c^2\) [Evaluate the power]
25 = \(c^2\) [Add]
5 = c [Take the positive square root of each side]
Hence, the distance between shop A and shop C is 5 miles.
The Pythagorean Theorem can be applied to right angled triangles only.
The Pythagorean theorem can be used to find the slope or steepness of mountains and hills. This theorem is also very useful in the fields of engineering, navigation and construction.
The converse of the Pythagorean theorem states that if the sum of squares of shorter sides of a triangle is equal to the square of its longest side then the triangle is right angled.
If \(a^2~+~b^2~=~c^2\) , then a, b and c are known as Pythagorean triples.
The hypotenuse of a right triangle is the side that is always located opposite to the 90 degree angle in the triangle. So just by observing the side opposite to the right angle we can identify the hypotenuse.