Pythagoras Theorem Proof
Consider a right-angled triangle having sides A, B, and C. Here, AC is the longest side (hypotenuse), and AB and BC are the legs of the triangle. Draw a perpendicular line BD at AC as shown in the figure below,
In △ABD and △ACB,
∠A = ∠A (Common angle)
∠ADB = ∠ABC (90°)
Therefore, we can say △ABD ∼ △ ACB (By AA Similarity)
Similarly, △BDC ∼ △ACB
Hence, AD/AB = AB/AC
AB2 = AD × AC ⇢ (1)
And, CD/BC = BC/AC
BC2 = CD × AC ⇢ (2)
Adding equations (1) and (2),
AB2 + BC2 = AC × AD + AC × CD
AB2 + BC2 = AC (AD + CD)
AB2 + BC2 = AC × AC
AB2 + BC2 = AC2
Also, AC2 = AB2 + BC2
Hence proved.
Pythagoras Theorem | Formula, Proof and Examples
Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The theorem can be expressed as r2 = p2 + q2, where ‘r’ is the hypotenuse and ‘p’ and ‘q’ are the two legs often called perpendicular and base of the triangle.
Pythagoras Theorem explains the relationship between the three sides of a right-angled triangle and helps us find the length of a missing side if the other two sides are known. It is also known as the Pythagorean theorem.
In this article, we will learn about the Pythagoras theorem statement, its formula, proof, examples, applications, and converse of Pythagoras theorem in detail.