Question

define and prove Pythagoras theorem​

Answer 1

Answer:

The proof of Pythagorean Theorem in mathematics is very important. In a right angle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. States that in a right triangle that, the square of a (a2) plus the square of b (b2) is equal to the square of c (c2).

Step-by-step explanation:

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Answer 2

Answer:

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 the base, perpendicular, and hypotenuse.

According to Pythagoras theorem,

(AC)2=(AB)2 + (BC)2

Proof:

Given, a triangle ABC in which ∠ABC is 900.

Construction: Draw a perpendicular BD on AC i.e. BD ⊥ AC. In ΔABD and ΔABC  we have,

∠BAD = ∠BAC i.e. ∠A is common in both triangles.

∠ABC = ∠ADB = 900

Therefore ΔABC∼ΔABD ( By AA similarity i.e. angle-angle similarity)

So,⇒ADAB=ABAC⇒AB2 = AD×AC ...(1)

In ΔBDC and ΔABC  we have,

∠BCD = ∠BCA i.e. ∠C is common in both triangles.

∠ABC = ∠ADC = 900

Therefore ΔABC∼ΔBDC ( By AA similarity i.e. angle-angle similarity)

So,⇒DCBC=BCAC⇒BC2 = AC×DC ...(2)

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

⇒AB2 + BC2 = AD×AC + AC× DC⇒AB2 + BC2 = AC(AD + DC)⇒AB2 + BC2 = AC(AC)⇒AB2 + BC2 = AC2

Hence, proved.  

Note: In a right-angled triangle, the hypotenuse is the longest side of the triangle and is opposite to the right angle i.e. 900. By drawing a perpendicular from point B and dividing the triangle ABC into 2 parts and using angle-angle similarity to prove the Pythagoras theorem.

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