The Pythagorean Theorem, also described as the ‘Pythagoras theorem,’ is arguably the most famous formula in mathematics that specifies the relationships between the sides of a ideal triangle.

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The theorem is attributed to a Greek mathematician and also thinker called Pythagoras (569-500 B.C.E.). He has actually many contributions to math, yet the Pythagorean Theorem is the a lot of vital of them.

Pythagoras is attributed with a number of contributions in math, astronomy, music, faith, philosophy, and so on One of his remarkable contributions to math is the discovery of the Pythagorean Theorem. Pythagoras studied the sides of a best triangle and uncovered that the amount of the square of the two shorter sides of the triangles is equal to the square of the longest side.

This article will certainly discuss what the Pythagorean Theorem is, its converse, and also the Pythagorean Theorem formula. Before getting deeper right into the topic, let’s respeak to the right triangle. A best triangle is a triangle with one inner angle equals 90 degrees. In a ideal triangle, the two short legs satisfy at an angle of 90 degrees. The hypotenuse of a triangle is oppowebsite the 90-level angle.

## What is the Pythagorean Theorem?

The Pythagoras theorem is a mathematical law that says that the sum of squares of the lengths of the two short sides of the best triangle is equal to the square of the length of the hypotenuse.

The Pythagoras theorem is algebraically composed as:

a2 + b2 = c2  They are drawn in such a way that they form a right triangle. We can create their areas have the right to in equation form:

Area of Square III = Area of Square I + Area of Square II

Let’s expect the length of square I, square II, and square III are a, b and also c, respectively.

Then,

Area of Square I = a 2

Area of Square II = b 2

Area of Square III = c 2

Hence, we deserve to write it as:

a 2 + b 2 = c 2

which is a Pythagorean Theorem.

## The Converse of the Pythagorean Theorem

The converse of the Pythagorean theorem is a preeminence that is supplied to classify triangles as either right triangle, acute triangle, or obtuse triangle.

Given the Pythagorean Theorem, a2 + b2 = c2, then:

For an acute triangle, c22 + b2, where c is the side opposite the acute angle.For a right triangle, c2= a2 + b2, where c is the side of the 90-degree angle.For an obtusage triangle, c2> a2 + b2, wbelow c is the side opposite the obtuse angle.

Example 1

Classify a triangle whose dimensions are; a = 5 m, b = 7 m and c = 9 m.

Solution

According to the Pythagorean Theorem, a2 + b2 = c2 then;

a2 + b2 = 52 + 72 = 25 + 49 = 74

But, c2 = 92 = 81Compare: 81 > 74

Hence, c2 > a2 + b2 (obtuse triangle).

Example 2

Classify a triangle whose side lengths a, b, c, are 8 mm, 15 mm, and also 17 mm, respectively.

Solutiona2 + b2 = 82 + 152 = 64 + 225 = 289But, c2 = 172 = 289Compare:289 = 289

As such, c2 = a2 + b2 (ideal triangle).

Example 3

Classify a triangle whose side lengths are given as;11 in, 13 in, and also 17 in.

Solutiona2 + b2 = 112 + 132 = 121 + 169 = 290c2 = 172 = 289Compare: 289 2 2 + b2 (acute triangle)

## The Pythagoras Theorem Formula

The Pythagoras Theorem formula is offered as:

⇒ c2 = a2 + b2

where;

c = Length of the hypotenuse;

a = size of one side;

b = length of the second side.

We can use this formula to deal with assorted troubles including right-angled triangles. For circumstances, we deserve to use the formula to determine the third size of a triangle when the lengths of two sides of the triangle are known.

### Application of Pythagoras Theorem formula in Real Life

We have the right to usage the Pythagoras theorem to examine whether a triangle is a ideal triangle or not.In oceanography, the formula is offered to calculate the rate of sound waves in water.Pythagoras theorem is offered in meteorology and aeroarea to identify the sound source and also its variety.We can use the Pythagoras theorem to calculate digital components such as tv displays, computer system display screens, solar panels, etc.We deserve to use the Pythagorean Theorem to calculate the gradient of a particular landscape.In navigation, the theorem is offered to calculate the shortest distance between given points.In architecture and building and construction, we deserve to usage the Pythagorean theorem to calculate the slope of a roof, drainage system, dam, and so on.

Worked examples of Pythagoras theorem:

Example 4

The two brief sides of a right triangle are 5 cm and 12cm. Find the size of the 3rd side

Solution

Given, a = 5 cm

b = 12 cm

c = ?

From the Pythagoras Theorem formula; c2 = a2 + b2, we have;

c2 = a2 + b2

c2 =122 + 52

c2 = 144 + 25

√c2 = √169

c = 13.

Because of this, the 3rd is equal to 13 cm.

Example 5

The diagonal and one side length of a triangular side is 25cm and 24cm, respectively. What is the measurement of the 3rd side?

Solution

Using Pythagoras Theorem,

c2 = a2 + b2.

Let b = third side

252 = 242 + b2625 = 576 + b2625 – 576 = 576 – 576 + b249 = b2b 2 = 49

b = √49 = 7 cm

Example 6

Find the dimension of a computer display screen whose dimensions are 8 inches and 14 inches.

Hint: The diagonal of the display screen is its size.

Solution

The dimension of a computer display screen is the same as the diagonal of the screen.

Using Pythagoras Theorem,

c2 = 82 + 152

Solve for c.

c2 = 64 + 225

c2 = 289

c = √289

c = 17

Hence, the dimension of the computer system display is 17 inches.

Example 7

Find the best triangle area provided that the diagonal and also the bases are 8.5 cm and also 7.7 cm, respectively.

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Solution

Using Pythagoras Theorem,

8.52 = a2 + 7.52

Solve for a.

72.25 = a2 + 56.25

72.25 – 56.25 = k2 + 56.25 – 56.25

16 = a2

a = √16 = 4 cm

Area of a appropriate triangle = (½) x base x height

= (½ x 7.7 x 4) cm2

= 15.4 cm2

Practice Questions

A 20 m lengthy rope is extended from the top of a 12 m tree to the ground. What is the distance between the tree and the finish of the rope on the ground?A 13 m lengthy ladder is leaning versus the wall. If the ground distance in between the foot of the ladder and the wall is 5 m, what is the wall’s height?