2. Pythagoras theorem
The Pythagorean theorem is related to the study
of sides of a right angled triangle.
It is also called as Pythagoras theorem.
The Pythagorean theorem states that,
In a right triangle
(length of the hypotenuse)2 =
{(1st side)2 + (2nd side)2}
a
b
cc2 = a2 + b2
3. In a right angled triangle three sides: Hypotenuse,
Perpendicular and Base. The base and the perpendicular
make an angle is 900.So, according to Pythagorean
theorem:
Pythagoras theorem
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
Pythagoras Theorem Proof:
p
b
h
A
B C
Given: Δ ABC is a right angled triangle where <B = 900
And AB = P, BC= b and AC = h
To Prove: h2 = p2 + b2
4. Construction : Put a perpendicular BD from B to AC ,
where AD = x and CB = h-x ,
p
b
h
A
B
C
D
x
(h-x)
Proof : Consider the two triangles
Δ ABC and Δ ABD, these two triangles
are similar to each other because of
AA similarity. This is because both the
triangle have a right angle and one
common angle at A.
In Δ ABC and Δ BDC both are similar
So by these similarity,
= =
p
h
x
p
b
h
(h-x)
b
AND
Pythagoras theorem
5. = =
p
h
x
p
b
h
(h-x)
b
AND
P2 = h * x and b2 = h (h – x)
Adding both L.H.S. and R.H. S.
P2 + b2 = h.x + h (h - x)
Or P2 + b2 = h.x + h2 – h.x
h2 = p2 + b2
Pythagoras theorem
p
b
h
A
B
C
D
x
(h-x)
6. converse of the Pythagoras Theorem
In a triangle, if square of one side is equal to the sum of the
squares of the other two sides, then the angle opposite the
first side is a right angle.
A
B C
Given: In a triangle ABC in which AC2 = AB2 + BC2
To prove: ∠ B = 90
7. Construction: A Δ PQR right angled at Q such that
PQ = AB and QR = BC (figure)
Proof: from Δ PQR,
we have: PR2 = PQ2 + QR2
(Pythagoras Theorem, as ∠ Q = 90 )
or, PR2 = AB2 + BC2
(By construction) ……………. (i)
converse of the Pythagoras Theorem
P
Q R
A
B C
8. But given, AC2 = AB2 + BC2……………………. (ii)
From (i) and (ii)
AC = PR …………………………….(iii)
Now, in Δ ABC and Δ PQR,
AB = PQ …………………(By construction)
BC = QR ………………...(By construction)
AC = PR …………….Proved in (iii) above
P
Q R
A
B C
converse of the Pythagoras Theorem
9. So, Δ ABC ≅ Δ PQR …………….(SSS congruence)
Therefore, ∠ B = ∠ Q ...........................(CPCT)
But, ∠ Q = 90 …………………..(By construction)
So, ∠ B = 90 Proved
P
Q R
A
B C
converse of the Pythagoras Theorem
10. The End
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