Question

State and prove Pythagoras theorem !!!!!

Answer 1

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STATEMENT :-

• Pythagoras theorem states that “ In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”. The sides of the right-angled triangle are called base, perpendicular and hypotenuse . Proof: Given, a triangle ABC in which ∠ABC is 900.

PROOF :-

We know, △ADB ~ △ABC

Therefore, AD/AB = AB/AC (corresponding sides of similar triangles)

Or, AB2 = AD × AC ……………………………..……..(1)

Also, △BDC ~△ABC

Therefore, CD/BC = BC/AC (corresponding sides of similar triangles)

Or, BC2= CD × AC ……………………………………..(2)

Adding the equations (1) and (2) we get,

AB2 + BC2 = AD × AC + CD × AC

AB2 + BC2 = AC (AD + CD)

Since, AD + CD = AC

Therefore, AC2 = AB2 + BC2

Hence, the Pythagorean theorem is proved.

Answer 2

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Statement: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Given: ABC is a triangle in which ∠ABC=90

∘

Construction: Draw BD⊥AC.

Proof:

In △ADB and △ABC

∠A=∠A [Common angle]

∠ADB=∠ABC [Each 90

∘

]

△ADB∼△ABC [A−A Criteria]

So,

AB/AD=AC/AB

Now, AB ^2=AD×AC ..........(1)

Similarly,

BC ^2=CD×AC ..........(2)

Adding equations (1) and (2) we get,

AB^2 +BC^2 =AD×AC+CD×AC

=AC(AD+CD)

=AC×AC

∴AB^2 +BC^2 =AC^2

[henceproved]

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