Physics, asked by huhishehdh, 1 year ago

Prove converse of Pythagoras theorem!

Answers

Answered by Anonymous
2

\huge\underline\mathfrak\purple{Statement}

In a triangle, if the square of one side is equal to the sum of square of other two sides then prove that the triangle is right angled triangle.

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\huge\underline\mathfrak\purple{Solution}

Given : AC² = AB² + BC²

To prove : ABC is a right angled triangle.

Construction : Draw a right angled triangle PQR such that, angle Q = 90°, AB = PQ, BC = QR.

Proof : In triangle PQR,

Angle Q = 90° ( by construction )

Also,

PR² = PQ² + QR² ( By using Pythagoras theorem )...(1)

But,

AC² = AB² + BC² ( Given )

Also, AB = PQ and BC = QR ( by construction )

Therefore,

AC² = PQ²+ QR²....(2)

From eq (1) and (2),

PR² = AC²

So, PR = AC

Now,

In ∆ABC and ∆PQR,

AB = PQ ( By construction )

BC = QR ( By construction )

AC = PR ( Proved above )

Hence,

∆ABC is congruent to ∆PQR by SSS criteria.

Therefore, Angle B = Angle Q ( By CPCT )

But,

Angle Q = 90° ( By construction )

Therefore,

Angle B = 90°

Thus, ABC is a right angled triangle with Angle B = 90°

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Hence proved!

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Answered by Anonymous
3

Answer:

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."

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 c2 = a2 + b2, then triangle must be right angle.

Pythagorean Theorem Test

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

Converse of Pythagorean Theorem

In △EGF,

By Pythagoras Theorem

EF2 = EG2 + FG2 = b2 + a2 ............(1)

In △EGF,

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 converse of Pythagorean theorem also hold.

Hence Proved.

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