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

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The to organize is attributed to a Greek mathematician and also philosopher named Pythagoras (569-500 B.C.E.). He has plenty of contributions to mathematics, however the Pythagorean to organize is the most crucial of them.

Pythagoras is credited with numerous contributions in mathematics, astronomy, music, religion, philosophy, etc. One of his significant contributions to mathematics is the discovery of the Pythagorean Theorem. Pythagoras studied the sides of a right triangle and also discovered the the sum of the square of the two much shorter sides of the triangles is equal to the square of the longest side.

This article will talk about what the Pythagorean to organize is, that converse, and the Pythagorean to organize formula. Prior to getting deeper into the topic, let’s recall the ideal triangle. A ideal triangle is a triangle v one inner angle equals 90 degrees. In a appropriate triangle, the two brief legs satisfy at an edge of 90 degrees. The hypotenuse that a triangle is the opposite the 90-degree angle.

What is the Pythagorean Theorem?

The Pythagoras organize is a mathematical law that claims that the amount of squares that the lengths of the two short sides of the best triangle is same to the square that the length of the hypotenuse.

The Pythagoras to organize is algebraically composed as:

a2 + b2 = c2

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They are drawn in together a way that they kind a ideal triangle. We have the right to write their areas can in equation form:

Area that Square III = Area that Square I + Area that Square II

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

Then,

Area that Square I = a 2

Area of Square II = b 2

Area of Square III = c 2

Hence, we can write that 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 rule that is offered to share triangles together either appropriate triangle, acute triangle, or obtuse triangle.

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

For an acute triangle, c22 + b2, wherein c is the next opposite the acute angle.For a ideal triangle, c2= a2 + b2, whereby c is the side of the 90-degree angle.For one obtuse triangle, c2> a2 + b2, whereby c is the next opposite the obtuse angle.

Example 1

Classify a triangle who 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 next lengths a, b, c, space 8 mm, 15 mm, and 17 mm, respectively.

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

Therefore, c2 = a2 + b2 (right triangle).

Example 3

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

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

The Pythagoras theorem Formula

The Pythagoras organize formula is given as:

⇒ c2 = a2 + b2

where;

c = size of the hypotenuse;

a = length of one side;

b = size of the 2nd side.

We have the right to use this formula to deal with various difficulties involving right-angled triangles. For instance, we have the right to use the formula to determine the 3rd length the a triangle as soon as the lengths of two sides of the triangle are known.

Application that Pythagoras theorem formula in actual Life

We deserve to use the Pythagoras to organize to check whether a triangle is a best triangle or not.In oceanography, the formula is supplied to calculate the speed of sound waves in water.Pythagoras to organize is supplied in meteorology and also aerospace to identify the sound source and that range.We can use the Pythagoras organize to calculate electronic contents such together tv screens, computer screens, solar panels, etc.We deserve to use the Pythagorean organize to calculation the gradient the a details landscape.In navigation, the theorem is provided to calculation the shortest distance between given points.In architecture and also construction, we deserve to use the Pythagorean theorem to calculate the slope of a roof, drainage system, dam, etc.

Worked examples of Pythagoras theorem:

Example 4

The two short sides the a ideal triangle room 5 cm and 12cm. Find the size of the 3rd side

Solution

Given, a = 5 cm

b = 12 cm

c = ?

From the Pythagoras to organize formula; c2 = a2 + b2, us have;

c2 = a2 + b2

c2 =122 + 52

c2 = 144 + 25

√c2 = √169

c = 13.

Therefore, the third is equal to 13 cm.

Example 5

The diagonal and also one side length of a triangular next is 25cm and 24cm, respectively. What is the dimension of the third side?

Solution

Using Pythagoras Theorem,

c2 = a2 + b2.

Let b = 3rd 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 screen who dimensions room 8 inches and 14 inches.

Hint: The diagonal line of the screen is that is size.

Solution

The dimension of a computer system screen is the exact same as the diagonal line of the screen.

Using Pythagoras Theorem,

c2 = 82 + 152

Solve for c.

c2 = 64 + 225

c2 = 289

c = √289

c = 17

Hence, the size of the computer screen is 17 inches.

Example 7

Find the ideal triangle area offered that the diagonal and also the bases room 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 the a right triangle = (½) x base x height

= (½ x 7.7 x 4) cm2

= 15.4 cm2

Practice Questions

A 20 m lengthy rope is stretched from the optimal of a 12 m tree come the ground. What is the distance in between the tree and the finish of the rope ~ above the ground?A 13 m lengthy ladder is leaning against the wall. If the soil distance between the foot of the ladder and the wall is 5 m, what is the wall’s height?