What is Pythagoras theorem ?
Answers
Step-by-step explanation:
c=a2+b2
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 triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°
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 a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.
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 the Pythagorean Theorem. The formula and proof of this theorem are explained here with examples.
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 triangle 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 a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.
Pythagoras Theorem-Right Angle Triangle
History
The theorem is named after a greek Mathematician called Pythagoras.
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.
Pythagoras Theorem
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
Example
The examples of theorem based on the statement given for right triangles is given below:
Consider a right triangle, given below:
Pythagoras theorem example
Find the value of x.
X is the side opposite to right angle, hence it is a hypotenuse.
Now, by the theorem we know;
Hypotenuse2 = Base2 + Perpendicular2
x2 = 82 + 62
x2 = 64+36 = 100
x = √100 = 10
Therefore, we found the value of hypotenuse here.
Pythagoras Theorem Proof
Given: A right-angled triangle ABC, right-angled at B.
To Prove- AC2 = AB2 + BC2
Construction: Draw a perpendicular BD meeting AC at D.
Pythagoras theorem Proof
Proof:
We know, △ADB ~ △ABC
Therefore, ADAB=ABAC (corresponding sides of similar triangles)
Or, AB2 = AD × AC ……………………………..……..(1)
Also, △BDC ~△ABC
Therefore, CDBC=BCAC (corresponding sides of similar triangles)
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 theorem 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.
Useful For
Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.
How to use?
To use this theorem, remember the formula given below:
c2 = a2 + b2
Where a, b and c are the sides of the right triangle.
For example, if the value of a = 3 cm, b = 4 cm, then find the value of c.
We know,
c2 = a2 + b2
c2 = 32+42
c2 = 9+16
c2 = 25
c = √25
c = 5
Hence, the third side is 5 cm.
As we can see, a + b > c
3 + 4 > 5
7 > 5
Hence, c = 5 cm is the hypotenuse of the given triangle.
How to find whether a triangle is a right-angled triangle?
If we are provided with the length of three sides of a triangle, then to find whether the triangle is a right-angled triangle or not, we need to use the Pythagorean theorem.
Let us understand this statement with the help of an example.
Suppose a triangle with sides 10, 24, and 26 are given.
Clearly, 26 is the longest side.
It also satisfies the condition, 10 + 24 > 26
We know,
c2 = a2 + b2 ………(1)
So, let a = 10,b = 24 and c = 26
First we will solve R.H.S. of equation 1.
a2 + b2 = 102 + 242 = 100 + 576 = 676
Now, taking L.H.S, we get;
c2 = 262 = 676
We can see,
LHS = RHS
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
Let,
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 angle opposite to the 13 unit side will be at a right angle.
Pythagoras theorem-problem 1
Problem 2: The two sides of a right-angled triangle are given as shown in the figure. Find the third side.
Pythagoras theorem-problem 2
Solution: Given;
Perpendicular = 15 cm
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 cm
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
Pythagoras Theorem-problem 3
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
ac = 4√2.
Thus, the length of the diagonal is 4√2 cm.