Proof of converse of Pythagoras theorem with state ment!.......❤️

1.	Proof of converse of Pythagoras theorem with state ment!.......❤️
Answer» $\huge\underline\mathscr{Statement}$ 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. ________________________ $\huge\underline\mathscr{Solution}$ 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° ____________________ Hence proved!

Answer»

$\huge\underline\mathscr{Statement}$

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.

________________________

$\huge\underline\mathscr{Solution}$

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°

____________________

Hence proved!

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