Question

Hey Guys explain full Pythagoras' theorem (Any nonsense type of answers will be reported​

Answer 1

Answer:

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:

the square of the hypotenuse is equal to

the sum of the squares of the other two side

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Answer 2

Answer:

Pythagoras Theorem Statement

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangles have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say x, y and z) which has positive integer values, when squared are put into an equation, also called a Pythagorean triple.

Right Angle Triangle

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.

Pythagoras Theorem

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

Pythagoras Theorem Proof

Given: A right-angled triangle ABC.

To Prove- AC2 = AB2 + BC2

Pythagoras Theorem Proof

Proof: First, we have to drop a perpendicular BD onto the side AC

We know, △ADB ~ △ABC

Therefore, ADAB=ABAC (Condition for similarity)

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

Also, △BDC ~△ABC

Therefore, CDBC=BCAC (Condition for similarity)

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 thoerem is proved.

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

39,12,310

Applications of Pythagoras Theorem

To know if the triangle is a right-angled triangle or not.

In a right-angled triangle, we can calculate the length of any side if the other two sides are given.

To find the diagonal of a square.

Pythagorean Theorem Problems

Problem 1: The sides of a triangle are 5,12 & 13 units. Check if it has a right angle or not.Pythagoras Theorem - Example

Solution: From Pythagoras Theorem, we have;

Perpendicular2 + Base2 = Hypotenuse2

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units

122 + 52 = 132

⇒ 144 + 25 = 169

⇒ 169 = 169

L.H.S. = R.H.S.

Therefore, the angles opposite to the 13 unit side will be at a right angle.

Problem 2: The two sides of a right-angled as shown in the figure. Find the third side.

Pythagoras Theorem Problem

Solution: Given;

Perpendicular = 15cm

Base = b cm

Hypotenuse = 17 cm

As per the Pythagorean Theorem, we have;

Perpendicular2 + Base2 = Hypotenuse2

⇒152 + b2 = 172

⇒225 + b2 = 289

⇒b2 = 289 – 225

⇒b2 = 64

⇒b = √64

Therefore, b = 8

Problem 3: Given the side of a square to be 4 cm. Find the length of the diagonal.

Solution- Given;

Sides of a square = 4 cm

Pythagoras Theorem Question

To Find- The length of diagonal ac.

Consider triangle abc (or can also be acd)

(ab)2 +(bc)2 = (ac)2

(4)2 +(4)2= (ac)2

16 + 16 = (ac)2

32 = (ac)2

(ac)2 = 32

or ac = 4√2.

Thus, the length of the diagonal is 4√2.

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Frequently Asked Questions on Pythagoras Theorem

What is the formula for Pythagorean Theorem?

The formula for Pythagoras, for a right-angled triangle, is given by; c2=a2+b2

What is the formula for hypotenuse?

The hypotenuse is the longest side of the right-angled triangle, opposite to right angle, which is adjacent to base and perpendicular. Let base, perpendicular and hypotenuse are a, b and c respectively. Then the hypotenuse formula, from the Pythagoras statement, will be;

c = √(a2 + b2)

Can we apply the Pythagoras Theorem for any triangle?

No, this theorem is applicable only for the right-angled triangle.

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