Solved Examples on Pythagoras Theorem

Let’s solve some questions on Pythagoras Theorem.

Example 1: In the below given right triangle, find the value of y.

Solution: 

By the statement of the Pythagoras theorem we get,

⇒ z² = x² + y²

Now, substituting the values directly we get,

⇒ 13² = 5² + y²

⇒ 169 = 25 + y²

⇒ y² = 144

⇒ y = √144 = 12

Example 2: Given a rectangle with a length of 4 cm and breadth of 3 cm. Find the length of the diagonal of the rectangle.

Solution: 

In the above diagram length of the rectangle is 4 cm, and the width is 3 cm. Now we have to find the distance between point A to point C or point B to point D. Both give us the same answer because opposite sides are of the same length i.e., AC = BD. Now let’s find the distance between points A and C by drawing an imaginary line.

Now triangle ACD is a right triangle. 

So by the statement of Pythagoras theorem,

⇒ AC² = AD² + CD²

⇒ AC² = 4² + 3²

⇒ AC² = 25

⇒ AC = √25 = 5

Therefore length of the diagonal of given rectangle is 5 cm.

Example 3: The sides of a triangle are 5, 12, and 13. Check whether the given triangle is a right triangle or not.

Solution: 

Given,

⇒ a = 5

⇒ b = 12

⇒ c = 13

By using the converse of Pythagorean Theorem,

⇒ a² + b² = c²

Substitute the given values in the above equation,

⇒ 13² = 5² + 12²

⇒ 169 = 25 + 144

⇒ 169 = 169

So, the given lengths satisfy the above condition.

Therefore, the given triangle is a right triangle.

Example 4: The side of a triangle is of lengths 9 cm, 11 cm, and 6 cm. Is this triangle a right triangle? If so, which side is the hypotenuse?

Solution: 

We know that hypotenuse is the longest side. If 9 cm, 11 cm, and 6 cm are the lengths of the angled triangle, then 11 cm will be the hypotenuse.

Using the converse of Pythagoras theorem, we get

⇒ (11)² = (9)² + (6)²

⇒ 121 = 81 + 36

⇒ 121 ≠ 117

Since, both the sides are not equal therefore 9 cm, 11 cm and 6 cm are not the side of the right-angled triangle.

Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The theorem can be expressed as r² = p² + q², where ‘r’ is the hypotenuse and ‘p’ and ‘q’ are the two legs often called perpendicular and base of the triangle.

Pythagoras Theorem explains the relationship between the three sides of a right-angled triangle and helps us find the length of a missing side if the other two sides are known. It is also known as the Pythagorean theorem.

In this article, we will learn about the Pythagoras theorem statement, its formula, proof, examples, applications, and converse of Pythagoras theorem in detail.