Pythagorean Theorem

When the triangle is at the right angle of 90 degrees and squares are drawn on all the three sides, then the biggest square has the same area calculation compared to other two squares when put together.

This is known as Pythagorean Theorem which can be written as the following:

a^2 + b^2 = c^2

Keep a note that c is the longest side of the triangle (hypotenuse); a and b are the two other sides.

hypotenuse

The mathematical definition of Pythagorean Theorem

In any right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two other sides.

Suppose the triangle PQR has measures of ‘5,12,13’ and has a right angle in it.

pythagorean

Let us verify–

52 + 122 = 132
25 + 144 = 169
169 = 169 (verified)

If we have the measure of any two sides of the right angle triangle, then we can find out the remaining length of the triangle by this theorem of Pythagorean. Please note that this only works for a right angle triangle.

Equation evaluation

Now we will write it as the equation of triangle ABC

AB2 + BC2 = AC2

Then we can use algebra for finding out the missing value in these examples.

Now let us start with examples.

Example 1: Find out the third side by this theorem.

theorem

Starting with this statement – AB2 + BC2 = AC2. Now we know that two sides are 3 and 4. Then we suppose that AB = 3 and BC = 4.

AC2 = AB2 + BC2
AC2 = 32 + 42
AC2 = 25

Thus, AC = 5. (According to the Pythagorean Theorem)

Example – 2: Find out the other side of AB where AC and BC are 13 and 5 respectively.

example 2.1

For solving, start with AB2 + BC2 = AC2, where we need to find out AB.

Thus, AB2 = AC2 – BC2
AB2 = 132 – 52
AB2 = 144

Thus, AB = 12 (as per the Pythagorean Theorem)

Example – 3: Check out whether this triangle is the right angle or not with the triangle having three sides values 26, 24, and 10.

example 3.1

First check out AB2 + BC2 = AC2

AB2 + BC2 = 102 + 242 = 676
AC2 = 676

Thus AC = 26

Thus, this proves that it is a right angle triangle as it is proved via the Pythagorean Theorem.

Thus, we here understood the concept of Pythagorean Theorem, with the example and detailed understanding of the concept. If this is clear, you will feel confident when solving related examples.

Pythagorean Theorem Triplets (Memorize These!)

Some Pythagorean triplets are best memorized. Knowing these will save you a lot of time. Below are the common triplets:

  • 3,4,5
  • 6,8,10
  • 5,12,13
  • 10, 24, 26
  • 8,15,17
  • 7,24,25