One of the cornerstones of geometry, the Pythagorean Theorem describes the relation between any right triangle's sides. The theorem says that in any right triangle, the square of the length of the hypotenuse-the side opposite the right angle is equal to the sum of the squares of the lengths of the other two sides. This is mathematically given as, where c represents the hypotenuse, a and b are the two other sides. The present theorem holds a great deal of significance not only in

theoretical mathematics but also in current-day applications in many areas. For

example, it might be used in construction in order to have the right angles in the building structures correct, in navigation to determine the shortest distance between any two points, and also in computer graphics where distances have to be calculated in drawing objects.

The theorem is simple yet a powerful tool for the solution of problems, which can be expressed in the form of a right triangle; therefore, it is fundamental to geometric understanding and problem-solving. Its use goes beyond abstract exercises; it influences and forms actual applications and technological inventions.

Formula For Pythagorean Theorem

The formula is given as,

where,

c is the hypotenuse
a and b are the lengths of the other two sides

Pythagorean Theorem Derivation

In the given figure below, consider the right-angled triangle ΔABC.

Let BD is perpendicular to the side AC.

The corresponding figure has been drawn from which the following ΔABC and ΔADB can be considered,

In ΔABC and ΔADB,

∠ABC = ∠ADB = 90°

Also, ∠A = ∠A → common

Using the AA criterion for the similarity of triangles,

ΔABC ~ ΔADB

Therefore, AD/AB = AB/AC

⇒ AB2 = AC x AD ……(1)

From the above figure, we consider ΔABC and ΔBDC.

∠C = ∠C → common

∠CDB = ∠ABC = 90°

Using the Angle Angle(A) criterion for the similarity of triangles, we conclude that,

ΔBDC ~ ΔABC

Therefore, CD/BC = BC/AC

⇒ BC2 = AC x CD ….(2)

From the similarity of triangles, we conclude that,

∠ADB = ∠CDB = 90°

So, if a perpendicular is drawn from the right-angled vertex of a right triangle to the hypotenuse, then the triangles formed on both sides of the perpendicular are similar to each other and also to the whole triangle.

To Prove: AC2 =AB2 +BC2

By adding equation (1) and equation (2), we get:

AB2 + BC2= (AC x AD) + (AC x CD)

AB2 + BC2 = AC (AD + CD) ….(3)

As AD+CD=AC put this value in equation (3).

AB2+BC2= AC (AC)

Now it will be written as

AB2+BC2= AC2

Thus the theorem is proved.

Applications of Pythagoras Theorem in Real Life

Pythagoras theorem is used in checking whether any given triangle is a right angle or not.
Through the Pythagoras theorem, aerospace scientists and meteorologists find the range and source of sound
Oceanographers utilize the Pythagoras theorem in calculating the speed of sound of water

Examples & Solutions of Pythagorean Theorem

Question 1: Determine the hypotenuse of a triangle, where two sides are given as 4 cm and 10 cm.

Solution: Applying the Pythagoras theorem,

Pythagorean Theorem Formula

Thus, the hypotenuse of the triangle is 10.77 cm.

Question 2: The hypotenuse of a right-angled triangle measures 13 cm and one of the sides measures 5 cm. Find the third side.

Solution: Here,

Hypotenuse a = 13 cm

One side b = 5 cm

Another side c = ?

Using the Pythagoras theorem,

Pythagorean Theorem Formula

c = 12 cm

Therefore, the other side of the triangle is 12 cm.

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