The Pythagoras 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. Explore the formula, derivation, and applications here.
The Pythagoras theorem is one of the most fundamental and essential principles in geometry. Named after the ancient Greek philosopher and mathematician Pythagoras, it remains a cornerstone of mathematical study.
Pythagoras Theorem Class 10
The Pythagoras theorem illustrates the relationship between the three sides of a right-angled triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the base and the perpendicular. This theorem allows us to calculate unknown side lengths and verify triangle properties. In this guide, we will explore the formula, mathematical derivations, and step-by-step solved examples. Let’s get started!
Pythagoras Theorem Formula Class 10
Pythagoras theorem formula: To understand the theorem, let’s review the properties of a right-angled triangle. A right triangle contains one 90° angle. Its sides are categorized as the Hypotenuse, Base, and Perpendicular. The hypotenuse is the longest side, located directly opposite the 90° angle, while the two remaining inner angles must sum to 90 degrees.
According to the Pythagoras theorem, the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. This is expressed as the Pythagorean equation, defining the relationship between sides a, b, and the hypotenuse c.
Perpendicular2 + Base2 = Hypotenuse2
| a² + b² = c² |
Consider a triangle ABC, where BC² = AB² + AC². In this equation, AB represents the base, AC represents the perpendicular, and BC represents the hypotenuse.
State Pythagoras Theorem
The theorem is attributed to the Greek mathematician Pythagoras of Samos. Born around 569 BC, he traveled extensively, including to Egypt, to advance his study of mathematics. He established a school of thinkers who emphasized mathematical rigor, and the theorem was named in his honor after he provided a formal proof.
Pythagoras Theorem Proof for Class 10
There are various ways to prove the Pythagoras Theorem. The two most common approaches are the algebraic method and the method involving similar triangles. We will examine both approaches in detail.
Pythagoras Theorem Proof using Algebraic Method
The algebraic proof of the Pythagorean Theorem uses a geometric construction: a large square containing four identical right-angled triangles and a smaller inner square with sides equal to the hypotenuses of those triangles.
The legs of the right triangles have lengths 'a' and 'b', and the hypotenuse is 'c'. The large square has sides of length (a + b), meaning its total area is (a + b)².
Since the inner square has side length 'c', its area is c². The total area of the large square is equal to the sum of the four triangles plus the inner square. Therefore:
(a+b)² = 4(½ × a × b) + c²
a² + b² + 2ab = 2ab + c²
a² + b² = c² (Proved)
Pythagoras Theorem Proof using similar triangles
Two triangles are considered similar if their corresponding angles are equal and their sides are in proportion. Let’s explore how to prove the Pythagoras theorem using the properties of similar triangles.
- In a right triangle, the hypotenuse is divided into two pieces by an altitude taken from the right-angled vertex
- The two smaller right triangles that resulted are similar to one another.
- Measure the three triangles’ three sides collectively. The corresponding sides of the three triangles are proportional to one another because they are comparable triangles.
- Consider the ratios of the corresponding sides. From the equations, get the squares of the original triangle’s sides.
- Use the squares of the sides to prove the Pythagorean Theorem. Then, determine whether the sum is equal to the square of the hypotenuse.
Pythagoras Theorem- Pythagorean triples Explained
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the equation a² + b² = c². In these triples, 'c' represents the hypotenuse, while 'a' and 'b' represent the base and perpendicular. The most famous example is the (3, 4, 5) triple.
| Pythagorean Triples Table | |||
| (3, 4, 5) | (5, 12, 13) | (8, 15, 17) | (7, 24, 25) |
| (20, 21, 29) | (12, 35, 37) | (9, 40, 41) | (28, 45, 53) |
| (11, 60, 61) | (16, 63, 65) | (33, 56, 65) | (48, 55, 73) |
| (13, 84, 85) | (36, 77, 85) | (39, 80, 89) | (65, 72, 97) |
Pythagoras Theorem Derivation of Hypotenuse Formula
Consider a triangle ABC, right-angled at B. Draw a perpendicular line segment BD that meets the hypotenuse AC at point D. We can use the similarity of the resulting triangles to prove the theorem.
In triangles ΔABD and ΔACB:
∠A = ∠A (Common angle)
∠ABC = ∠ADB (Both are right angles)
Therefore, ΔABD ∼ ΔACB (by the AA similarity criterion)
Similarly, we can demonstrate that ΔBCD ∼ ΔACB.
Since ΔABD ∼ ΔACB,
Hence, AD/AB = AB/AC, which leads to AD × AC = AB².
Similarly, we can show that ΔBCD ∼ ΔACB.
Hence, CD/BC = BC/AC, which implies CD × AC = BC².
By combining these two equations, we obtain:
AB² + BC² = (AD × AC) + (CD × AC)
AB² + BC² = AC(AD + DC)
AB² + BC² = AC² [Proved]
Pythagoras Theorem Model
In summary, the Pythagoras Theorem proves that the square of the hypotenuse is equal to the sum of the squares of the perpendicular and the base.
This theorem is mathematically expressed as the relationship between the three sides of a right triangle: a, b, and hypotenuse c.
Perpendicular2 + Base2 = Hypotenuse2
| a² + b² = c² |
Pythagoras Theorem Image
Pythagoras Theorem Calculator
The primary application of the Pythagoras theorem is calculating side lengths and angles in right-angled triangles. Beyond academic mathematics, it has significant real-world applications in engineering, construction, and navigation.
Perpendicular2 + Base2 = Hypotenuse2
Or simply: a² + b² = c², where 'a' and 'b' are the legs and 'c' is the hypotenuse.
Pythagoras Theorem Applications
Here are several examples of how the Pythagoras Theorem is applied:
- The Pythagorean Theorem is frequently used to determine the lengths of a right-angled triangle’s sides. We can get the formula for the base, perpendicular, and hypotenuse using the Pythagoras Theorem.
- To determine the diagonal length of a rectangle, square, etc., use the theorem.
- A triangle’s correctness is determined using the converse of Pythagoras Theorem.
- In trigonometry, Pythagoras Theorem is used to determine trigonometric ratios such as sin, cos, tan, cosec, sec, and cot.
- In the disciplines of engineering and building, architects apply the Pythagoras Theorem approach.
- Applying the Pythagoras Theorem to mountain surveying
- Finding the shortest path is another application for it in navigation.
- Calculating how steep a mountain or hill’s slope requires the theorem.
Pythagoras Theorem Based Questions
Q. A triangle has legs of 4 cm and 3 cm. What is the length of the hypotenuse?
Solution: Using the formula Perpendicular² + Base² = Hypotenuse²
In this problem, Base = 4 cm, Perpendicular = 3 cm.
Applying the theorem:
Hypotenuse² = 4² + 3²
Hypotenuse² = 16 + 9 = 25
Hypotenuse = √25 = 5 cm
Q. Given a triangle with sides 5 cm, 13 cm, and 12 cm, determine if it is a right-angled triangle.
Solution: A triangle is right-angled if it satisfies the Pythagoras theorem.
We must verify if 5² + 12² = 13².
L.H.S = (5² + 12²) = (25 + 144) = 169
R.H.S = (13²) = 169
L.H.S = R.H.S
Since the equality holds, the triangle is indeed a right-angled triangle.
Q. A triangle has a hypotenuse of 17 cm and a base of 8 cm. Find the perpendicular.
Solution:
Base (a) = 8 cm, Hypotenuse (c) = 17 cm. Let the perpendicular be 'b'. Using b² = c² - a², we find b² = 17² - 8² = 289 - 64 = 225. Therefore, b = √225 = 15 cm.
Pythagoras Theorem Class 10 worksheet
Q1. A ladder is placed 12 cm from a wall, and its top reaches 5 cm up the wall. Find the length of the ladder.
Q2. Does a triangle with sides 9 cm, 40 cm, and 41 cm form a right-angled triangle?
Q3. A triangle has sides of 16 cm and 63 cm. What is the length of the hypotenuse?
FAQs
The Pythagoras theorem is expressed as Perpendicular² + Base² = Hypotenuse², or a² + b² = c².
There are several proofs for the Pythagoras Theorem, including algebraic methods and similarity-based proofs, both covered in this article.
No, the Pythagorean theorem is exclusive to right-angled triangles.
Some problems may be complex, but once you master the foundational concept of the Pythagoras theorem, you can solve most associated questions with ease.
Any triangle containing a 90° angle satisfies the Pythagoras theorem, where the square of the hypotenuse (AC²) equals the sum of the squares of the legs (AB² + BC²).
The side opposite the 90° angle (the hypotenuse) is always the longest side, as it is opposite the largest angle in the triangle.