Pythagoras’ theorem

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.

Example

Finding the Length of the Hypotenuse:

Find the length x.

x is the length of the hypotenuse corresponding to the value ‘c’ in the formula

c² = a² + b².

c² = 84² + 13².

= 7056 + 169

= 7225

C = √7225 = 85

Hence the length of the hypotenuse = 85cm.

Example

Find the length of the unknown leg in the adjoining diagram.

We can take a = x, b = 48 and c = 50.
a² = c² – b².
x² = 50² – 48².
= 2500 – 2304 = 196
x = √196 = 14
Hence the length of the leg = 14″.

Pythagoras in real life

Pythagorean theorem is used to find the lengths, distances and heights using right triangles which model real life situations.

When fire occurs in high raise buildings, the fire fighting men cannot use the regular stairs or lifts. They can reach some floors using ladders. In order to determine the ladder length, they apply the Pythagorean theorem as they can estimate the height of the floor affected and the horizontal distance they can use to keep the ladder in position.
Example

A ladder of length 16 ft is placed against a wall. If the foot of the ladder is 7 ft from the wall find the height the ladder reaches on the wall nearest to the tenth of a foot.

Solution

The situation is described using the right triangle ABC, where AB represents the wall, BC the floor and AC the ladder.
It is required to find the measure h, which is the length of a leg in right triangle ABC. Using Pythagorean theorem,

h² = AC² – BC².
= 16² – 7² = 256 – 49 = 207.

h = √207 = 14.4 ft (rounded to the tenth of a foot).    

Assessment

A ladder of length 20 ft is placed against a wall. If the foot of the ladder is 10 ft from the wall find the height the ladder reaches on the wall nearest to the tenth of a foot.