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Pythagoras Theorem

Pythagoras Theorem - Class 10

The “Pythagorean theorem,” was inspired by the ancient Greek Mathematician Pythagoras who invented the theorem during 500 B.C. It has been argued that the “Ancient Babylonians” already understood the theorem long before the invention by Pythagoras. They knew the relationship between the sides of the triangle and while solving for the hypotenuse of an isosceles triangle, they came up with the approximate value of up to 5 decimal places.

What is Pythagoras theorem?

In a right-angled triangle, “The sum of squares of the lengths of the two sides is equal to the square of the length of the hypotenuse (or the longest side).”

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

In the given triangle, side “a” is called as “Perpendicular”, side “b” is called as “Base” and side “c” is called as “Hypotenuse.”

The side opposite to the right angle ( $90^{\circ}$ ) is the longest side known as Hypotenuse, as we know that the side opposite to the greatest angle is longest.

Note- Pythagoras theorem is only applicable to a Right-Angled triangle.

Another way of representation of the Pythagoras Theorem-

Another way of Pythagoras Theorem proof is described here. Consider a right-angled triangle having perpendicular as a, base as b, and hypotenuse as c. Consider three squares of sides a,b,c mounted on the three sides of a triangle having the same sides as shown.

By Pythagoras Theorem –

Area of square A + Area of square B = Area of square C

Example- Prove the Pythagoras Theorem for a right angle triangle having sides to be 3cm, 4cm and 5 cm.Solution –

From Pythagoras Theorem we have,

$(P e r p e n d i c u l a r)^{2} + (b a s e)^{2} = (H y p o t e n u s e)^{2}$

Perpendicular = 3 cm

Base = 4 cm

Hypotenuse = 5 cm

$(3)^{2} + (4)^{2} = (5)^{2}$

$\Rightarrow 9 + 16 = 25$

$\Rightarrow 25 = 25$

L.H.S. = R.H.S.

Therefore Pythagoras theorem is proved.

Pythagoras Theorem - Class 10

Proof of Pythagoras Theorem-

To Prove- $A C^{2} = A B^{2} + B C^{2}$ </p >

Pythagoras Theorem - Class 10

For this we drop a perpendicular BD onto the side AC

We know, $△ A D B \sim△ A B C$

Therefore, $\frac{A D}{A B} = \frac{A B}{A C}$ (Condition for similarity)

Or, $A B^{2} = A D \times A C$ ……..(1)

Also, $△ B D C \sim△ A B C$

Therefore, $\frac{C D}{B C} = \frac{B C}{A C}$ (Condition for similarity)

Or, $B C^{2} = C D \times A C$ ……..(2)

Adding the equations (1) and (2) we get,

$A B^{2} + B C^{2} = A D \times A C + C D \times A C$

$A B^{2} + B C^{2} = A C (A D + C D)$

Since, AD + CD = AC

Therefore, $A C^{2} = A B^{2} + B C^{2}$

Application-

(1st): To know the triangle is right-angled or not. It is to be noted that if the Pythagoras Theorem is proved then it must be a right-angled triangle.

Example- The sides of a triangle are 5,12 & 13 units. Check if it has a right angle or not.Solution-

To prove- Pythagoras theorem in order to find whether it has a right angle or not

From Pythagoras Theorem, we have-

$(P e r p e n d i c u l a r)^{2} + (b a s e)^{2} = (H y p o t e n u s e)^{2}$

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units

$(12)^{2} + (5)^{2} = (13)^{2}$

$\Rightarrow 144 + 25 = 169$

$\Rightarrow 169 = 169$

L.H.S. = R.H.S.

Therefore the angles opposite to the 13 unit side will be at a right angle. Pythagoras Theorem - Class 10

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FAQs on Pythagoras Theorem - Class 10

1. What is Pythagoras Theorem?

Ans. Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. It can be represented as a^2 + b^2 = c^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

2. How is Pythagoras Theorem used in real life?

Ans. Pythagoras Theorem is used in various real-life applications. It is commonly used in construction and architecture to ensure the accuracy of right angles and to calculate the dimensions of structures. It is also used in navigation, such as determining the distance between two points on a map or finding the shortest route between two locations.

3. How can Pythagoras Theorem be proved?

Ans. Pythagoras Theorem can be proved using different methods, such as algebraic, geometric, or trigonometric proofs. One of the most common geometric proofs involves constructing squares on each side of the right-angled triangle and showing that the areas of the squares satisfy the theorem's relationship.

4. Can Pythagoras Theorem be applied to non-right-angled triangles?

Ans. No, Pythagoras Theorem is only applicable to right-angled triangles. In non-right-angled triangles, the relationship between the sides is determined by the Law of Cosines or the Law of Sines, depending on the available information.

5. What are some practical examples of Pythagoras Theorem?

Ans. Pythagoras Theorem can be applied to various practical situations. For example, it can be used to calculate the diagonal length of a rectangular field, the height of a ladder required to reach a certain height on a wall, or the distance between two points on a coordinate plane.

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Dec 19, 2024 Last updated

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