Pythagoras Theorem StatementPythagoras theorem
asserts that In a right-angled triangle, the square of the hypotenuse side
is equivalent to the total of squares of the other two sides . The sides of
this triangle have been defined as Perpendicular, Base and Hypotenuse. Here,
the Hypotenuse is the longest side since it is against the
angle of 90 . The sides of a right triangle (assume a, b and c), which have
positive integer values while squared, are put into an equalization, also
called a Pythagorean triple.
HistoryThe theory is named
after a Greek Mathematician named Pythagoras. Pythagoras Theorem FormulaSuppose there is a
triangle with a -
perpendicular, b - the base, c - Hypotenuse.
Hypotenuse2
= Perpendicular2 + Base2 c2 = a2
+ b2 The side contrary
to the right angle (90 ) is the longest side (also known as Hypotenuse) since
the side contrary to the greatest angle is the largest.Consider 3 squares
of sides a, b, c positioned on the three sides of a triangle holding the same
sides as presented.With Pythagoras
Theorem square A (Area)+
square B (Area) = square C. (Area)
ExampleThe cases of
theorem and based on the observation given for right triangles is given below:Consider a right
triangle, given below:Find the value of
x.Since X is the side
opposite to right angle, it is a hypotenuse.Now, by the theorem
we knowHypotenuse
2
= Base
2+ Perpendicular
2x
2=
8
2+ 6
2x
2=
64+36 = 100x = 100 = 10Therefore, the
value of x is 10. Proof Pythagoras TheoremGiven: A right-angled
triangle ABC, right-angled at B.Prove: AC
2 = AB
2
+ BC
2Construction: Form a
perpendicular BD joining AC at D.Proof:We identify ADB ~ ABCTherefore,
ADAB=ABAC (corresponding
surfaces of similar triangles)Or, AB
2
= AD AC .. ..(1)Also, BDC ~ ABCTherefore,
CDBC=BCAC (corresponding
sides of similar triangles)Or, BC
2=
CD AC ..(2)Adding the
equations (1) and (2) we get,AB
2
+ BC
2 = AD AC + CD ACAB
2
+ BC
2 = AC (AD + CD)Since, AD + CD = ACTherefore, AC
2=
AB
2 + BC
2Hence, the
Pythagorean theorem is proved.
Note: Pythagorean
theorem is only applicable to the Right-Angled triangle. Applications of Pythagoras Theorem
To check if the triangle is a right-angled
triangle or not.
For every right-angled triangle, we can find
the length of any side if the other two sides are given.
To find the diagonal of a square.
Useful ForPythagoras theorem
is helpful to determine the sides of a right-angled triangle. Suppose if we
know the two sides of a right triangle, then we can find the third side.
How to use it?To use this
theorem, recollect the formula given below:c
2
= a
2+ b
2(Let the right
triangle sides be a, b and c)For instance, if
the value of a = 6 cm, b = 8 cm, then find the value of c.We know, c
2=
a
2 + b
2c
2
=6
2+8
2c
2
= 36 + 64c
2
= 100c = 10Hence, the 3rd side
is 5 cm.As we can see, a + b > c6 +8 > 1014 > 10Hence the
hypotenuse is c = 10 cm
How to check whether a triangle is a right-angled triangle?Let us explain this
statement with the help of an example.Assume a triangle
with sides 20, 48, and 52 are given. 52 is the longest
side.It also satisfies
the condition, 20 + 48 > 52We know,c
2
= a
2 + b
2 (1)So, let a = 20, b =
48 and c = 52First we will find R.H.S. of equation 1.a
2
+ b
2= 20
2 + 48
2 = 400 + 2304
= 2704Now, taking L.H.S,
we getc
2=
52
2 = 2704We can see, LHS = RHSHence, the above
triangle is a right triangle Pythagorean Theorem Problems
Ques 1: The sides of a
triangle are 15, 36 39 units.
Solution: From Pythagoras
Theorem, we have
Perpendicular2
+ Base2 = Hypotenuse2Let,Perpendicular = 36
unitsBase = 15 unitsHypotenuse = 39
units {since it is the longest side measure}36
2
+ 15
2 = 39
2 1296+ 225 = 1521 1521 = 1521 L.H.S. = R.H.S.Therefore, the
angle opposite to the 39 units side will be a right angle.
Problem 2: The 2 sides of a
right-angled triangle are given. Find the third side.
Solution: GivenPerpendicular
= 12 cmBase = b cmHypotenuse = 13 cmAs per the
Pythagorean Theorem, we have
Perpendicular2
+ Base2= Hypotenuse2 12
2+ b
2 =13
2 225 + b
2= 289 b
2 = 289 225 b
2= 64 b = 64Therefore, b = 8 cm
Problem 3: Assume the
side of a square to be 3 cm. Find the length of the diagonal
.Solution- GivenSides of a square =
3 cmTo get- The length
of diagonal ac.Suppose triangle
abc (or can also be acd)
2
+(bc) 2 = (ac) 2
2
+(3) 2= (ac) 2
2
2
c = (a2
+ b2)
This resource was uploaded by: Reshmi
Other articles by this author