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In a right-angled triangle, “The sum of squares of the lengths of the two sides is equal to the square of the length of the hypotenuse (or the longest side).”


What is Pythagoras theorem - Class 7

a2+b2=c2

a^2+b^2=c^2


In the given triangle, side “a” is called as “Perpendicular”, side “b” is called as “Base” and side “c” is called as “Hypotenuse.”

The side opposite to the right angle (90) is the longest side known as Hypotenuse, as we know that the side opposite to the greatest angle is longest.

Note- Pythagoras theorem is only applicable to a Right-Angled triangle.

Another way of representation of the Pythagoras Theorem-

Another way of Pythagoras Theorem proof is described here. Consider a right-angled triangle having perpendicular as a, base as b, and hypotenuse as c. Consider three squares of sides a,b,c mounted on the three sides of a triangle having the same sides as shown.


What is Pythagoras theorem - Class 7By Pythagoras Theorem –


Area of square A + Area of square B = Area of square C

Example- Prove the Pythagoras Theorem for a right angle triangle having sides to be 3cm, 4cm and 5 cm.Solution –

From Pythagoras Theorem we have,

(Perpendicular)2+(base)2=(Hypotenuse)2

Perpendicular = 3 cm

Base = 4 cm

Hypotenuse = 5 cm

(3)2+(4)2=(5)2

⇒9+16=25

⇒25=25

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

Therefore Pythagoras theorem is proved.

What is Pythagoras theorem - Class 7

 

Proof of Pythagoras Theorem-To Prove- AC2=AB2+BC2</p >What is Pythagoras theorem - Class 7For this we drop a perpendicular BD onto the side ACWe know, △ADB∼△ABCTherefore, ADAB=ABAC (Condition for similarity)Or, AB2=AD×AC……..(1)Also, △BDC∼△ABCTherefore, CDBC=BCAC (Condition for similarity)Or, BC2=CD×AC……..(2)Adding the equations (1) and (2) we get,AB2+BC2=AD×AC+CD×ACAB2+BC2=AC(AD+CD)Since, AD + CD = ACTherefore, AC2=AB2+BC2Application-(1st): To know the triangle is right-angled or not. It is to be noted that if the Pythagoras Theorem is proved then it must be a right-angled triangle.

Example- The sides of a triangle are 5,12 & 13 units. Check if it has a right angle or not.Solution-

To prove- Pythagoras theorem in order to find whether it has a right angle or not

From Pythagoras Theorem, we have-

(Perpendicular)2+(base)2=(Hypotenuse)2

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units

(12)2+(5)2=(13)2

⇒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.

What is Pythagoras theorem - Class 7
(2nd): In a right-angled triangle, we can calculate the length of any side if other two sides are given.

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


What is Pythagoras theorem - Class 7 


Solution-

Given-

Perpendicular = 15cm

Base = b cm

Hypotenuse = 17 cm

From Pythagoras Theorem, we have

(Perpendicular)2+(base)2=(Hypotenuse)2

152+b2=172

⇒225+b2=289

⇒b2=289–225

⇒b2=64

⇒b=64−−√

Therefore b=8

(3rd): To find the diagonal of a square

Example- Given the side of a square to be 4 cm. Find the length of the diagonal.Solution- 

Given –

Sides of a square = 4 cm

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 acl=42–√What is Pythagoras theorem - Class 7

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