Question

state and prove pythagoras Theorem​

Answer 1

Answer:

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Answer 2

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History

The theorem is named after a greek Mathematician called Pythagoras.

Pythagoras Theorem Formula

Consider the triangle given above:

Where “a” is the perpendicular side,

“b” is the base,

“c” is the hypotenuse side.

According to the definition, the Pythagoras Theorem formula is given as:

Hypotenuse2 = Perpendicular2 + Base2

c2 = a2 + b2

The side opposite to the right angle (90°) is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.

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

The examples of theorem based on the statement given for right triangles is given below:

Consider a right triangle, given below:Find the value of x.

X is the side opposite to right angle, hence it is a hypotenuse.

Now, by the theorem we know;

Hypotenuse2 = Base2 + Perpendicular2

x2 = 82 + 62

x2 = 64+36 = 100

x = √100 = 10

Therefore, we found the value of hypotenuse here.Given: A right-angled triangle ABC, right-angled at B.

To Prove- AC2 = AB2 + BC2

Construction: Draw a perpendicular BD meeting AC at D.

Proof:

We know, △ADB ~ △ABC

Therefore, ADAB=ABAC (corresponding sides of similar triangles)

Or, AB2 = AD × AC ……………………………..……..(1)

Also, △BDC ~△ABC

Therefore, CDBC=BCAC (corresponding sides of similar triangles)

Or, BC2= CD × AC ……………………………………..(2)

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

AB2 + BC2 = AD × AC + CD × AC

AB2 + BC2 = AC (AD + CD)

Since, AD + CD = AC

Therefore, AC2 = AB2 + BC2

Hence, the Pythagorean theorem is proved.

Note: Pythagorean theorem is only applicable to Right-Angled triangle.

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