Introduction
Pythagorean theorem which states the special relationship between the sides of a right triangle is perhaps the most popular and most applied theorem in Geometry. The algebraic statement of the Pythagorean theorem is used to derive the distance formula in coordinate Geometry and to prove the Pythagorean identities in Trigonometry. In fact, the fundamentals of Trigonometry are taught using the ratios of the sides of a right triangle.
Right triangles and Pythagorean theorem are not only used to solve real life problems, but often used in solving many advanced problems in Mathematics and Physical Sciences.
Euclid used squares drawn on the sides of the right angles and showed the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the legs of a right triangle.
The algebraic form of the statement of Pythagoras theorem c2 = a2 + b2 is used in solving right triangles.
The statement of the Pythagorean theorem is as follows:
In a right triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
If Δ ABC is a right triangle,
then the hypotenuse is the side opposite to the right angle (AB) and the legs (BC and CA) are the sides containing the right angle.
Then according to Pythagorean theorem
BC2 + CA2 = AB2 or
a2 + b2 = c2.
Pythagoras’ Theorem Formula
The algebraic form of the Pythagorean theorem
c2 = a2 + b2.
is used as a formula to solve for the third side of a right triangle if the lengths of any two sides are given.
If the lengths of the legs are given, the formula used to find the length of the hypotenuse is
c = √a2 + b2.
When the lengths of hypotenuse and one of the legs is known, we use one of the following formula to solve for the second leg.
a = √c2 – b2 or
b =√ c2 – a2
We will be using these formulas when solving few example problems on Pythagorean theorem…
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