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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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.

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

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.

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

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.

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.

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

**Step 1:**

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

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

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.

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