Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Given: We are given a right triangle ABC right angled at B.
To prove: AC²=AB²+BC²
Construction: Draw BD parallel to AC.
Proof: In triangles ADB and ABC, we have
Angle ADB = Angle ABC (Each 90⁰)
and, Angle A = Angle A (Common)
So, ΔABD is similar to ΔABC (AA similarity criterion)
⇒ AD/AB = AB/AC (In similar triangles corresponding sides are proportional)
⇒ AB²=AD*AC ---(1)
In triangle BDC and ABC, we have
Angle CDB = Angle ABC (Each 90⁰)
Angle C = angle C (Common)
So, ΔBDC is similar to ΔABC (In simialr triangles corresponding sides are proportional)
Adding equations 1 and 2, we get