⛄☺️Define: Converse of Pythagoras Theorem..? ❄️☃️
Answers
The converse of the Pythagorean Theorem is: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
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Answer:
Converse of Pythagoras theorem is defined as:
“If square of a side is equal to the sum of square of the other two sides then triangle must be right angle triangle”. Whereas Pythagorean theorem states that the sum of the square of two sides (legs) is equal to square of the hypotenuse of a right-angle triangle. But, in the reverse of the Pythagorean theorem, it is said that if this relation satisfies, then triangle must be right angle triangle. So, if the sides of a triangle have length, a, b and c and satisfy given condition a2 + b2 = c2, then the triangle is a right-angle triangle.
Converse of Pythagoras Theorem Proof
Let us assume the Pythagoras theorem is already proved.
Statement: If the length of a triangle is a, b and c and c² = a² + b² , then the triangle is a right-angle triangle.
example :
Proof: Construct another triangle, △EGF, such as AC = EG = b and BC = FG = a.
(see the picture)
In △EGF, by Pythagoras Theorem:
EF² = EG² + FG² = b² + a² …………(1)
In △ABC, by Pythagoras Theorem:
AB² = AC² + BC² = b² + a² …………(2)
From equation (1) and (2), we have;
EF² = AB²
EF = AB
⇒ △ ACB ≅ △EGF (By SSS postulate)
⇒ ∠G is right angle
Thus, △EGF is a right triangle.
Hence, we can say that the converse of Pythagorean theorem also holds.
Hence Proved.
Formula
As per the converse of the Pythagorean theorem, the formula for a right-angled triangle is given by:
a²+b² = c²
Where a, b and c are the sides of a triangle.
Applications
Basically, the converse of the Pythagoras theorem is used to find whether the measurements of a given triangle belong to the right triangle or not. If we come to know that the given sides belong to a right-angled triangle, it helps in the construction of such a triangle. Using the concept of the converse of Pythagoras theorem, one can determine if the given three sides form a Pythagorean triplet.
