Pythagoras Theorem
Pythagoras Theorem is an important topic in Maths, which explains the relation between the sides of a right-angled triangle. It is also sometimes called Pythagorean Theorem. The formula and proof of this theorem are explained here. This theorem is basically used for the right-angled triangle and by which we can derive base, perpendicular and hypotenuse formula.
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.
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.
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
Proof: First, we have to drop a perpendicular BD onto the side AC
We know, △ADB ~ △ABC
Therefore, \(\frac{AD}{AB}=\frac{AB}{AC}\) (Condition for similarity)
Or, AB2 = AD × AC ……………………………..……..(1)
Also, △BDC ~△ABC
Therefore, \(\frac{CD}{BC}=\frac{BC}{AC}\) (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.
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.
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.
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
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.
Frequently Asked Questions
What is the formula for Pythagorean Theorem?
The formula for Pythagoras, for a right-angled triangle, is given by;
Hypotenuse2 = Perpendicular2 + Base2
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.