Pythagoras Theorem | Formula, Proof and Examples

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.

It 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.

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

Pythagoras Theorem

In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

hypotenuse² = perpendicular² + base²

It provides us with the relationship between the sides in a right-angled triangle. A right triangle consists of two legs and a hypotenuse.

Pythagoras Theorem Formula

Pythagoras theorem formula is AC² = AB² + BC², where AB is the perpendicular side, BC is the base, and AC is the hypotenuse side. The Pythagoras equation is applied to any right-angled triangle, a triangle that has one of its angles equal to 90°.

The three sides of the right-angled triangle are called the Pythagoras Triplets. 

Pythagoras Theorem Proof

Consider a right-angled triangle having sides A, B, and C. Here, AC is the longest side (hypotenuse), and AB and BC are the legs of the triangle. Draw a perpendicular line BD at AC as shown in the figure below,

In △ABD and △ACB,

∠A = ∠A (Common angle)

∠ADB = ∠ABC (90°)

Therefore, we can say △ABD ∼ △ ACB (By AA Similarity)

Similarly, △BDC ∼ △ACB

Hence, AD/AB = AB/AC

AB² = AD × AC ⇢ (1)

And, CD/BC = BC/AC

BC² = CD × AC ⇢ (2)

Adding equations (1) and (2),

AB² + BC² = AC × AD + AC × CD

AB² + BC² = AC (AD + CD)

AB² + BC² = AC × AC

AB² + BC² = AC²

Also, AC² = AB² + BC²

Hence proved.

Converse of Pythagoras Theorem

The converse of the Pythagoras theorem states that,

Given a triangle with sides of length a, b, and c, if a² + b² = c², then the angle between sides a and b is a right angle. For any three positive real numbers a, b, and c such that a² + b² = c², there exists a triangle with sides a, b and c as a consequence of the converse of the triangle inequality.

Converse of Pythagoras Theorem Proof

For a triangle with the length of its sides a, b, if c² = a² + b², we need to prove that the triangle is right-angled.

We assume that it satisfies c² = a² + b², and by looking into the diagram, we can tell that ∠C = 90°, but to prove it, we require another triangle △EGF, such as AC = EG = b and BC = FG = a.

In △EGF, by Pythagoras Theorem:

⇒ EF² = EG² + FG2² = b² + a² ⇢ (1)

In △ABC, by Pythagoras Theorem:

⇒ AB² = AC² + BC² = b² + a² ⇢ (2)

From equation (1) and (2), we have;

⇒ EF² = AB²

⇒ EF = AB

⇒ △ ACB ≅ △EGF (By SSS postulate)

⇒ ∠G is right angle

Thus, △EGF is a right triangle. Hence, we can say that the converse of the Pythagorean theorem also holds.

History of Pythagoras Theorem

The history of the Pythagoras Theorem goes back to the ancient Babylon and Egypt eras. It is named after the ancient Greek mathematician and philosopher Pythagoras of Samos. He lived during the 6th century BCE.

But the roots of this theorem go to ancient cultures. It is very likely that Babylonians and Indians used this theorem well before Pythagoras, but its widespread use came into existence after Pythagoras stated it. The Pythagorean theorem is also known as the Baudhayana theorem, listed in the book Baudhāyana Śulbasûtra by the great Indian mathematician Baudhāyana.

One of the other reasons this theorem is known as Pythagoras or Pythagorean Theorem is because the disciples of Pythagoras spread knowledge and philosophy of Pythagoras after his death as well.

Pythagoras Theorem Applications

Below are some of the uses of Pythagorean Theorem in real life:

• Solving Right-Angled Triangles: Students use the theorem to calculate the length of unknown sides of right-angled triangles, given the other two sides. This is crucial for solving problems in trigonometry, mensuration, and coordinate geometry.
• Distance and Height Problems: The theorem is applied in real-world problems, such as finding the height of buildings or the distance between two points on a plane, helping students understand practical applications.
• Proofs and Derivations: Students also learn to prove Pythagoras’ Theorem and use it to derive other important mathematical properties, such as the distance formula in coordinate geometry.
• Vectors and 3D Geometry: Pythagoras' Theorem is applied in calculating the magnitude of vectors and distances between points in three-dimensional space.
• Calculus and Coordinate Geometry: The theorem underpins the derivation of formulas used in differential calculus and helps solve problems involving curves, tangents, and distances between geometric figures.
• Physics Applications: In physics, students use Pythagoras' Theorem to solve problems involving forces, velocities, and resultant vectors.
• Construction and Architecture: Builders and architects use Pythagoras' Theorem to ensure accurate measurements when constructing buildings, bridges, or other structures. It helps in creating right angles and calculating diagonal lengths when designing layouts and foundations.
• Navigation: In navigation, the theorem is used to calculate the shortest distance between two points on a map, which is essential for both air and sea travel. By treating distances as right-angled triangles, navigators can find the direct route between locations.

People Also Read:

• Angles in Right Angled Triangle
• Right Angled Triangle Formula
• Similar Triangles
• Congruence of Triangles

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.

Practice Questions on Pythagoras Theorem

Here are some practice questions on the Pythagoras theorem for you to solve.

Q1: If the two shorter sides of a right angled triangle measures 14 and 15 cm, find the length of the longest side.

Q2: If the hypotenuse and perpendicular of a right angled triangle are 5 and 4 cm then find the base.

Q3: In a triangular field the sides measures 24 cm, 7 cm and 25 cm then prove that field is the form of a right triangle.

Q4: A wall is of 12 m height and a ladder of 13 m is placed against it touching its top. Find the distance between the foot of the ladder and the wall.

Conclusion

Pythagorean Theorem is one of the most important concepts in mathematics. It helps us understand the relationship between the sides of a right-angled triangle and is widely used in various fields such as geometry, construction, physics, and even navigation. By knowing just two sides of a right-angled triangle, we can easily calculate the third, making it a powerful tool for solving real-world problems. This simple formula makes finding accurate solutions much easier in various everyday tasks, such as measuring objects, calculate distances etc.