For a right angled triangle
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Side opposite to right angle is Hypotenuse
Side adjacent to right angle are Base & height
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Note :
Here, we can also take AB as base & BC as height.
It does not change our answer
LetΒ Β AB = a
BC = b
AC = c
Pythagoras theorem says that
Β Square of hypotenuse = Sum of square of other two sides
Β Β a ^{ 2 } + b ^{ 2 } = c ^{ 2 }
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There are a lot of interesting things that we can do with Pythagoras theorem
Like
 Pythagoras triplets

Proof of Pythagoras theorem β The
normal way
Letβs do some questions
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Find x
Here,
Β βABC is a right angled triangle, right angled at B
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And,
Β AB = 3, BC = 4, AC = x
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By Pythagoras Theorem
AC ^{ 2 } = AB ^{ 2 } + BC ^{ 2 }
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x ^{ 2 } = 3 ^{ 2 } + 4 ^{ 2 }
x ^{ 2 } = 9 + 16
x ^{ 2 } = 25
x ^{ 2 } = 5 ^{ 2 }
Cancelling squares
Β x = 5
Β
Therefore, x = 5
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Find x
Here,
Β βABC is a right angled triangle, right angled at B
Β
And,
Β AB = 6 cm, BC = 8 cm, AC = x
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By Pythagoras Theorem
AC ^{ 2 } = AB ^{ 2 } + BC ^{ 2 }
Β
x ^{ 2 } = 6 ^{ 2 } + 8 ^{ 2 }
x ^{ 2 } = 36 + 64
x ^{ 2 } = 100
x ^{ 2 } = 10 ^{ 2 }
Cancelling squares
Β x = 10 cm
Β
Therefore, x = 10 cm
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Find x
Here,
Β βPQR is a right angled triangle, right angled at P
Β
And,
Β PR = 8 cm, PQ = 15 cm, QR = x
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By Pythagoras Theorem
QR ^{ 2 } = PR ^{ 2 } + PQ ^{ 2 }
^{ }
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Find x
Here,
Β βPQR is a right angled triangle, right angled at P
Β
And,
Β PQ = 24 cm, QR = x, PR = 7 cm
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By Pythagoras Theorem
QR ^{ 2 } = PQ ^{ 2 } + PR ^{ 2 }
Β Β Β Β Β x ^{ 2 } = (24) ^{ 2 } + 7 ^{ 2 }
Β Β Β Β Β x ^{ 2 } = 576 + 49
Β Β Β Β Β x ^{ 2 } = 625
Β Β Β Β Β x ^{ 2 } = (25) ^{ 2 }
Cancelling squares
Β x = 25 cm
Β
Therefore, x = 25 cm
625 = 5 Γ 5 Γ 5 Γ 5
Β Β Β Β = 25 Γ 25Β
Β Β Β Β = (25) ^{ 2 }
Β
Β
Β
Β
Find x
In βABC,
Β AB = AC
Hence, it is an isosceles triangle.
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Now, AD β₯ BC
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In isosceles triangle,
altitude and median are same
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β΄ AD is median of BC
i.e. D is midpoint of BC
β΄ BD = DC = BC/2
Β Β BD = DC = x/2
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In βADC, right angled at D.
By Pythagoras Theorem
(AB) ^{ 2 } = (AD) ^{ 2 } + (BD) ^{ 2 }
(37) ^{ 2 } + (12) ^{ 2 } + (x/2) ^{ 2 }
1369 = 144 + x ^{ 2 } /4
Β
1369 β 144 = x ^{ 2 } /4
1225 = x ^{ 2 } /4
Β Β 1225 Γ 4 = x ^{ 2 }
Β Β Β Β Β 4900 = x ^{ 2 }
Β Β Β Β Β Β Β x ^{ 2 } = 4900
Β
Β Β Β Β Β x ^{ 2 } =Β 70 ^{ 2 }
Cancelling squares
x = 70 cm
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β΄ x = 70 cm
Let's look at one last example
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