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One of the well-recognized formulas in modern mathematics is the Pythagorean Theorem, which renders us with the association between the sides in a right triangle. A right triangle has two legs and a hypotenuse. The two legs meet at an angle of 90Â° while the hypotenuse is the longest side of the right triangle and is that side which is opposite to the right angle. Simply, a Pythagoras equation describes the relationship between the three sides of a right-angled triangle.

The Pythagorean Theorem explains the link in every right triangle is:

aÂ² + bÂ² = cÂ²

The formula for Pythagoras Theorem is given by:

PerpendicularÂ² + BaseÂ² = HypotenuseÂ²

Or

aÂ² + bÂ² = cÂ²

Where a, b and c represents the sides of the right-angled triangle with hypotenuse as c.

The Pythagoras theorem is used to calculate the sides of a right-angled triangle. If we are given the lengths of two sides of a right-angled triangle, we can simply determine the length of the 3rd side. (Note that it only works for right-angled triangles!)

The theorem is frequently used in Trigonometry, where we apply trigonometric ratios such as sine, cos, tan; to find out the length of the sides of the right triangle.

Take into account a right-angled triangle Î”MNO. From the figure shown below, it is right-angled at N.

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Pythagorean Theorem Derivation - 1

Let NP be perpendicular to the side MO.

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Pythagoras Theorem Derivation - 2

From the above-given figure, consider the Î”MNO and Î”MPN,

In Î”MNO and Î”MPN,

âˆ MNO = âˆ MPN = 90Â°

âˆ M = âˆ M â†’ common

Using the MM criterion for the similarity of triangles we have,

Î” MNO ~ Î” MPN

Thus, MP/MN = MN/MO

â‡’ MNÂ² = MO x MPâ€¦(1)

Considering Î”MNO and Î”NPO from the figure below:

Pythagorean Theorem Derivation -3

âˆ O = âˆ OÂ â†’ common

Â âˆ OPN = âˆ MNO = 90Â°

Applying the principle of the Angle Angle(AA) criterion for the similarity of triangles, we come to the conclusion that,

Î”NPO ~ Î”MNO

Thus, OP/NO = NO/MO

â‡’ NOÂ² = MO x OP â€¦..(2)

From the similarity of triangles, we come to the conclusion that,

âˆ MPN = âˆ OPN = 90Â°

That said, if a perpendicular is constructed from the right triangle of a right-angled vertex to the hypotenuse, then the triangles so formed on both sides of the perpendicular are identical to each other and as well the whole triangle.

To Prove: MOÂ² = MNÂ² + NOÂ²

By adding up the equation (1) and equation (2), we obtain:

MNÂ² + NOÂ² = (MO x MP) + (MO x OP)

MNÂ² + NOÂ² = MO (MP + OP)â€¦.(3)

Since MP + OP = MO, substituting the value in equation (3).

MNÂ² + NOÂ² = MO (MO)

Now, it becomes

Â MNÂ² + NOÂ² = MOÂ²

Therefore, the Pythagorean theorem is proved.

Example:

Calculate the hypotenuse of a right-angled triangle whose lengths of two sides are 6 cm and 9 cm.

Solution: Given the criteria are:

Perpendicular = 9 cm

Base = 6 cm

Applying the Pythagoras theorem we have

HypotenuseÂ² = PerpendicularÂ² + BaseÂ²

Now, putting the values we have will get:

HypotenuseÂ² = 9Â² + 6Â²

HypotenuseÂ² = 81 + 36

Hypotenuse =âˆš117

HypotenuseÂ = âˆš10.8.

Example:

Solve the right-angled triangle with the two given sides 8, b, 17

Solution:

Begin with: aÂ² + bÂ² = cÂ²

Put in the values we know: 8Â² + bÂ² = 17Â² = 353

Calculate squares:Â 64 + bÂ² = 289

Take 64 from both sides: 64 âˆ’ 64 + bÂ² = 289 âˆ’ 64

Calculate: bÂ² = 225

Square root of both sides:Â b = âˆš225

Calculate: b = 15

Example:

Determine the distance of diagonal across a square of size 2?

Solution:

Begin with: aÂ² + bÂ² = cÂ²

Put in the values we know: 2Â² + 2Â² = cÂ²

Calculating the squares: 2 + 2 = cÂ²

2 + 2 = 4: Â 4 = cÂ²

Now, letâ€™s swap the sides:Â cÂ² = 4

Square root of both sides:Â c = âˆš4

This is about: 2.

FAQ (Frequently Asked Questions)

Q1. What is a Pythagorean Theorem?

Answer: The Pythagorean Theorem, also known as the Pythagoras theorem, implies that the square of the length of the hypotenuse is equivalent to the sum of squares of the lengths of other two sides angled at 90 degrees. In other words, the sum of the squares of the two legs of a right triangle is equivalent to the square of its hypotenuse.

Q2. Explain Some Real-Life Application of Pythagorean Theorem Formula?

Answer: The length of diagonal joining two buildings can be calculated using this Pythagoras theorem formula.

In addition, we use this formula accompanying the trigonometry concept, to evaluate the angle of elevation of an individual seeing an object placed at top of the building in regard to the horizontal line. The angle of depression can be found out when the object is placed below the line of sight of an individual.

Following are few other real-life applications of the Pythagoras theorem:

Verifies whether a given triangle is a right-angle triangle or not.

The Pythagorean theorem helps Meteorologists and Aerospace scientists in finding the range and sound source.

The formula for Pythagorean Theorem is used by oceanographers for identifying the speed of sound in water.

Q3. What is a Hypotenuse?

Answer: The hypotenuse is the longest side in a right-angled triangle. The square of the hypotenuse is equivalent to the sum of the squares of the other two remaining sides.