proof of Converse of Pythagoras theorem
Answers
Answer:
Converse of Pythagoras Theorem Proof
Statement: If the length of a triangle is a, b and c and c2 = a2 + b2, then the triangle is a right-angle triangle.
Converse of Pythagoras theorem
Proof: Construct another triangle, △EGF, such as AC = EG = b and BC = FG = a.
Converse of Pythagorean Theorem Proof
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.
Converse of Pythagoras Theorem Proof
Statement: If the length of a triangle is a, b and c and c2 = a2 + b2, then the triangle is a right-angle triangle.
Proof: Construct another triangle, △EGF, such as AC = EG = b and BC = FG = a.
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.
