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 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:

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

**Derivation:**

Consider a right angled triangle\Delta ABC. It is right angled at B.

Let BD be perpendicular to the side AC.

In \Delta ABC and \Delta ADB

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

\(\angle DBA =\angle BCA\)

\(\angle A =\angle A\)

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

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

\(\angle DAB =\angle CBD\)

\(\angle DBA =\angle BCD\)

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

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

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

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 \(AC^{2}=AB^{2}+BC^{2}\). We drop a perpendicular BD on the side AC.

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

∴ \(\frac{AD}{AB}=\frac{AB}{BC}\) (Condition for similarity) Or \(AB^{2}=AC\times AD\)…….. (1)

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

∴\(\frac{CD}{BC}=\frac{BC}{AC}\) (Condition for similarity) Or \(BC^{2}=AC\times CD\)……….. (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, \(CD+AD=AC\)

Thus, \(AC^{2}=AB^{2}+BC^{2}\)

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.
- 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.

**Solved Examples:**

**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

Other 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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