Pythagorean Theorem

Pythagorean Theorem in a right angled triangle states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides

 

(We do not intend to present here the proof of Pythagorean theorem, despite the fact that numerous proofs exist and the popularity that the proofs of this theorem enjoys. Rather, our aim is to apply pythagoras theorem to solve a few questions- a practical point of view)

Written as an equation, the Pythagorean theorem is

AB2 + BC2 = AC2

i.e.  a2 + b2 = c2

where a and b are the lengths of the two sides that contain the right angle, while c is the length of the side opposite the right angle.

The longest side in a right triangle is opposite of the right angle.

It is called hypotenuse.

The following right angled triangles are some examples illustrating the pythagoras theorem:

In triangle ABC, the Pythagorean theorem is

AB2 + BC2 = AC2

The lengths of the three sides AB, BC and AC are 3, 4 and 5.

Now, verify the pythagoras theorem by applying the values in the theorem’s statement as below:

32 + 42 = 52 

i.e. 9 + 16 = 25 which is a true statement.

Therefore, 3 and 4 are the legs of the right triangle, while 5 is hypotenuse length.

In the following table, some famous Pythagorean triplets are given


Legs of the Right triangle

Hypotenuse Length

pythagoras Theorem

6 and 8

10

62 + 82  = 102

5 and 12

13

52 + 122  = 132

7 and 24

25

72 + 242  = 252

Caution

If the lengths of two sides in a triangle are 3 and 4, it is not necessary that the third side must be 5 and therefore it is not necessary that a right triangle is definitely formed if two sides are 3 and 4. The third side need not necessarily be only 5. It can be any other number as the explanation below will clarify:

According to triangle inequality property of sides, in a triangle the length of any side should be less than the sum of the lengths of the other two sides in the triangle.

From this property, if ‘c’ is the length of the third side, then

c < 3 + 4, i.e. c < 7.

Any number less than 7 can be a value for c, the third side length, not necessarily 5.

But, if the lengths of the three sides of a triangle are 3, 4 and 5, then the triangle formed is certainly a right angled triangle; for,
32 + 42 = 52 

Converse of Pythagorean Theorem:

If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the angle opposite the longest side is a right angle.

Or briefly put, it is

If the square of one side equals the sum of the squares of the two other sides, then a right angle is formed.

In the triangle ABC below, the lengths of the sides are 10, 8 and 6. Does the triangle have a right angle?

Answer:

Check pythagoras theorem.

Is 102 = 62 + 82

Is 100 = 36 + 64?

Is 100 = 100?

Yes it is.

Therefore, a right angle is formed opposite the side of length 10

Two very important types of right angled triangle:

450–450-900 right angled triangle.

In the right angled triangle ABC above, the three angles are 450, 450 and 900.

If the length of the side

AB opposite 450 is x, and

BC opposite 450 is x, then

AC, the hypotenuse is of length x√2

300–600-900 right angled triangle

In the right angled triangle PQR above,

let the length of the side  

PQ opposite 300 be x; then

QR opposite 600 will be of length x√3; and

Hypotenuse PR will be of length 2x.