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Question

Explain the concept of the Pythagorean Theorem.


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Solution

Step 1: State the Pythagorean Theorem

In a right-angled triangle, the square of the length of the hypotenuse is the sum of the square of the base and the square of the perpendicular.

Thus, Hypotenuse2=Base2+Perpendicular2.

Step 2: Prove the Pythagorean Theorem

Consider the right-angled triangle, ABC, such that B=90.

Now, a perpendicular is drawn from the point B to the side CA.

It is known from the theorem that, The perpendicular drawn from the vertex of the right angle of a right-angled triangle to the hypotenuse then both sides of the perpendicular are similar to the whole triangle and to each other.

Therefore, ABD~ABC.

Thus, according to the condition of similarity, ADAB=ABAC.

AB2=AD×AC.....................1.

Therefore, DBC~ABC.

Thus, according to the condition of similarity, CDBC=BCAC.

BC2=CD×AC.....................2.

Step 3: Add equation (1) and (2).

AB2+BC2=AD×AC+CD×ACAB2+BC2=ACAD+CD

From the given figure it is clear that, AD+CD=AC.

AB2+BC2=AC×ACAB2+BC2=AC2AC2=AB2+BC2

Hence, it is proven that, (Hypotenuse)2=(Base)2+(Perpendicular)2.


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