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Converse of Pythagorean Theorem

Pythagorean theorem states that the sum of square of two sides (legs) is equal to square of hypotenuse of right angle triangle. But, in the converse of this theorem, it is said that if this relation satisfy then triangle must be right angle triangle. If the sides of triangle have length, a, b and c and satisfy given condition $a^2$ + $b^2$ = $c^2$, then the triangle must be a right angle triangle.

Converse of Pythagorean theorem is defined as that "If square of a side is equal to sum of square of other two sides then triangle must be right angle triangle."
In this section, we will see about converse of the Pythagorean Theorem and solving some problems in detail.

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Converse of Pythagorean Theorem Proof

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Before proving the converse of Pythagorean theorem, we have to assume that Pythagorean theorem is already proved.

If length of sides of triangles are a, b and c and $c^2$ = $a^2$ + $b^2$, then triangle must be right angle.

Pythagorean Theorem Test

Construct a another triangle, $\triangle$EGF, such as AC = EG = b and BC = FG = a.

Converse of Pythagorean Theorem

In $\triangle$EGF,

By Pythagoras Theorem

EF$^2$ = EG$^2$ + FG$^2$ = b$^2$ + a$^2$ ............(1)

In $\triangle$EGF,

By Pythagoras Theorem

AB$^2$ = AC$^2$ + BC$^2$ = b$^2$ + a$^2$ ............(2)

From equation (1) and (2), we have

EF$^2$ = AB$^2$

EF = AB

$\Rightarrow$ $\triangle$ ACB $\cong$ $\triangle$EGF (By SSS postulate)

$\Rightarrow$ $\angle$G is right angle

Thus, $\triangle$EGF is a right triangle.

Hence, we can say that converse of Pythagorean theorem also hold.

Hence Proved.

Applications

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Basically, the converse of Pythagoras theorem is used in order find whether or not a triangle is right triangle, when measures of its sides are given. If we come to know that the given sides belong to a right-angled triangle, then helps in construction of such triangle. Using the concept of converse of Pythagoras theorem, one can determine if the given three sides form a Pythagorean triplet. In short, converse of Pythagoras theorem is being utilized allover in the geometry.

Examples

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Given below are the problems based on converse of Pythagorean theorem.

Solved Examples

Question 1: The sides of a triangle are 5, 12 and 13. Check whether the given triangle is right triangle or not?
Solution:

Given:

a = 5

b = 12

c = 13

By using converse of Pythagorean Theorem,

$c^2$ = $a^2$ + $b^2$

Solving this, we need to substitute the given values in above equation,

$13^2$ = $5^2$ + $12^2$

169 = 25 + 144

169 = 169

So, it satisfies the above condition.

Therefore, the given triangle is a right triangle.



Question 2: The sides of a triangle are 7, 11 and 13. Check whether the given triangle is right triangle or not?
Solution:

Given:

a = 7

b = 11

c = 13

By using converse of Pythagorean Theorem,

$c^2$ = $a^2$ + $b^2$

Solving this, we need to substitute the given values in above equation,

$13^2$ = $7^2$ + $11^2$

169 = 49 + 121

169 = 170

So, it is not satisfied the above condition.

Therefore, the given triangle is not a right triangle.



Question 3: The sides of a triangle are 4,6 and 8. Say whether the given triangle is right triangle or not.
Solution:

Given: a = 4, b = 6, c = 8

Step 1:

The general form of Pythagorean theorem is,

$a^2$ + $b^2$ = $c^2$

Step 2:

$8^2$ = $4^2$ + $6^2$

64 = 16 + 36

64 = 52

Step 3:

The sides of given triangle are not satisfying the condition $a^2$ + $b^2$ = $c^2$. So, it is not satisfied the above condition.

Therefore, the given triangle is not a right triangle.



Practice Problems

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Practice with the following problems.
Problem 1 : Prove that the sides 7, 24 and 25 form a Pythagorean triplet.

Problem 2 : Check whether 5 cm, 12 cm and 16 cm are the sides of right triangle or not ?

Problem 3 : Find if $\sqrt{3}$, 5 and $\sqrt{2}$ make a right triangle ?

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