How to Find the Area of Right Angles Triangles using Pythagoras Theorem

The Pythagorean Theorem, also recognized as the “Pythagoras Theorem,” is by far the most well-known mathematical proof for defining the relationships between the sides of a right triangle. The great Pythagoras is known for his contributions to mathematics, astronomy, music, philosophy, and other subjects and fields. The Pythagorean Theorem is one of his most famous contributions to the world of education. Pythagoras investigated the sides of a right triangle and found that the number of the squares of the triangle’s two shorter sides equals the square of the triangle’s longest side.

Explanation of The Pythagorean Theorem:

A right-angled triangle is one in which any one of the three angles measures exactly 90 degrees. The hypotenuse of the triangle is the side that is opposite to the right angle, while the other sides are named as the base and height, respectively. Pythagoras’ Theorem describes the situations between a right angled triangle hypotenuse, base, and height.

Statement of Pythagorean Theorem:

The below stated is the Pythagorean theorem:

In any given respective right-angled triangle, the square of the hypotenuse side is invariably equivalent to the total sum of squares of the other two sides in the respective solved triangle.

This expresses that the area of the square on the hypotenuse of a right-angled triangle is invariably equal to the sum of areas of the squares on the opposite two sides of the triangle.

How can you prove the Pythagorean Theorem?

You can prove the Pythagorean Theorem in three fundamental ways:

  • By the use of Coordinate Geometry
  • By the use of Trigonometry
  • By the use of Similarity

Pythagoras Theorem Formula:

Let us imagine or consider the triangle ABC, Where “a” is the perpendicular of the triangle in this case, “b” is the base here, While “c” is the hypotenuse of the triangle. According to the definition stated, by Pythagoras the Pythagorean Theorem formula is given as:

Hypotenuse² = Perpendicular² + Base²

How to Know Whether a Triangle is a Right-angled Triangle?

If the students are provided with the length of all the three sides of a triangle, then to find out whether the respective triangle is a right-angled triangle or not, we will need to use the Pythagoras theorem and its formulas. Let us perfectly understand this theorem with the help of an easy and manageable example. Let us imagine a triangle with sides 10, 24, and 26 as respective. Here the 26 is the longest side of the triangle. Along with it, 26 also satisfies the condition, 10 + 24 > 26.

Now we know that, c² = a² + b² (equation 1). So, now let us name the sides as a = 10, b = 24 and c = 26. The first step has to be to solve R.H.S. of equation 1 which is given above.

a(square) + b(square) = 100 + 576 = 676. Then let us take L.H.S, and this gives us: c(square) = 262 = 676. We can thus prove that the left-hand side is equal to the right-hand side. LHS = RHS. Hence, the respective ABC triangle is a right triangle, as it satisfies and meets all the properties stated in the theorem.

Prove the Theorem at Home:

Get yourself a piece of paper, a pen, and a scissor, then study the below steps as a guide:

  • One can draw a right-angled triangle on a piece of paper, leaving plenty of space besides.
  • Then draw a square along the hypotenuse of the previously drawn triangle. 
  • Again draw the same sized square on the other side of the hypotenuse along with the same respective triangle. 
  • Then you can draw lines in the square drawn.
  • To make it apt cut out the shapes. 
  • Lastly, arrange them all so that one can prove that the big square has the same area as the two squares drawn on the side.
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