Pythagorean Theorem is one of the most fundamental theorems in mathematics and it defines the relationship between the three sides of a right-angled triangle. You are already aware of the definition and properties of a right-angled triangle. It is the triangle with one of its angles as a right angle, that is, 90 degrees. The side that is opposite to the 90-degree angle is known as the hypotenuse. The other two sides that are adjacent to the right angle are called legs of the triangle.

The Pythagoras theorem, also known as the Pythagorean theorem, states that the square of the length of the hypotenuse is equal to the sum of squares of the lengths of other two sides of the right-angled triangle. Or, the sum of the squares of the two legs of a right triangle is equal to the square of its hypotenuse.

Let us call one of the legs on which the triangle rests as its base. The side opposite to the right angle is its hypotenuse, as we already know. The remaining side is called the perpendicular. So, mathematically, we represent the Pythagoras theorem as:

HypotenuseÂ ^{2}Â Â =Â Perpendicular^{2}Â + Base^{2} |

## Pythagorean Theorem Derivation

Consider a right-angled triangle Î”ABC. It is right-angled at B.

Let BD be perpendicular to the side AC.

InÂ Î”ABC and Î”ADB

\(\angle ABC=\angle ABD = 90^{\circ}\) Â

\(\angle DBA =\angle BCA\)

\(\angle A =\angle A\)

Using the AA criterion for the similarity of triangles, \(\Delta ABC \cong \Delta ADB\)

Considering \(\Delta ABC and \Delta BDC\)

\(\angle DAB =\angle CBD\)

\(\angle DBA =\angle BCD\)

Â \(\angle CDB=\angle ADB=90^{\circ}\)

Using the AA criterion for the similarity of triangles, \(\Delta BDC\cong \Delta ABC\)

Thus, it can be concluded \(\Delta ADB\cong \Delta BDC\) Â Â

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.

Now, we are required to prove Â \(AC^{2}=AB^{2}+BC^{2}\). We drop a perpendicular BD on the side AC.

We already know that \(\Delta ADB\cong \Delta ABC\)

âˆ´ Â Â \(\frac{AD}{AB}=\frac{AB}{BC}\) (Condition for similarity) Or \(AB^{2}=AC\times AD\)â€¦â€¦.. (1)

Also, Â Â \(\Delta BDC\cong \Delta ABC\)

âˆ´\(\frac{CD}{BC}=\frac{BC}{AC}\) Â Â (Condition for similarity) Or \(BC^{2}=AC\times CD\)â€¦â€¦â€¦.. (2)

Summing equation (1) with equation (2),

\(AB^{2}+BC^{2}=CD.AC+AD.AC\)\(\Rightarrow AB^{2}+BC^{2}=AC\left ( CD+AD \right )\)

However, Â Â \(CD+AD=AC\)

Thus, Â Â \(AC^{2}=AB^{2}+BC^{2}\)

Hence, the proof of the Pythagoras theorem.

**Application of Pythagoras Theorem in Real Life****:**

- Pythagoras theorem is used to check if a given triangle is a right-angled triangle or not.Â
- Aerospace scientists and meteorologists find the range and sound source using the Pythagoras theorem.
- It is used by oceanographers to determine the speed of sound in water.

## Pythagorean Theorem Examples & Solutions

**Question 1: **Find the hypotenuse of a triangle whose lengths of two sides are 4 cm and 10cm.

**Solution: **Using the Pythagoras theorem,

\(Hypotenuse^{2}=Perpendicular^{2}+Base^{2}\)

\(Hypotenuse^{2}=10^{2}+4^{2}\)

\(Hypotenuse=\sqrt{10^{2}+4^{2}}=\sqrt{116}=10.77cm\)

Hence the hypotenuse of the triangle is 10.77 cm.

**Question 2: **If the hypotenuse of a right-angled triangle is 13 cm and one of the two sides is 5 cm, find the third side.

**Solution: **Given,

Hypotenuse a= 13 cm

One side b = 5 cm

Another side c=?

Using the Pythagoras theorem,

\(a^{2}=b^{2}+c^{2}\)

\(13^{2}=5^{2}+C^{2}\)

\(C^{2}=\sqrt{13^{2}-5^{2}}=\sqrt{169-25}=\sqrt{144}=12cm\)

Hence the hypotenuse of the triangle is 12cm.

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## Frequently Asked Questions on Pythagorean Theorem Formula

### What is the formula for Pythagoras Theorem?

The formula for Pythagoras Theorem is given by:

Hypotenuse^2 = Perpendicular^2 + Base^2

Or

c^2 = a^2+b^2

Where a,b and c are the sides of the right-angled triangle.

### What is meant by Pythagoras Theorem?

The Pythagoras theorem states that the square of the length of the hypotenuse is equal to the sum of squares of the lengths of other two sides of the right-angled triangle.

### What is the use of Pythagoras Theorem?

The Pythagoras theorem also known as Pythagorean theorem is used to find the sides of a right angled triangle. This theorem is mostly used in Trigonometry, where we use trigonometric ratios such as sine, cos, tan to find the length of the sides of the right triangle.

### What is the real life application of Pythagoras Theorem Formula?

The length of diagonal connecting two buildings can be calculated using this formula.

Also, we use this formula along with trigonometry concept, to find angle of elevation if a person seeing an object kept at top of the building with respect to horizontal line. Angle of depression is calculated when the object is kept below the line of sight of the person.