The Pythagorean Theorem, also referred to as the ‘Pythagoras theorem,’ is may be the most famous formula in mathematics that specifies the relationships between the sides of a best triangle.

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

Pythagoras is credited with several 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 political parties of a best triangle and also discovered that the amount of the square that the two much shorter sides of the triangle is equal to the square that the longest side.

This article will comment on what the Pythagorean organize is, the converse, and the Pythagorean to organize formula. Prior to getting deeper right into the topic, let’s recall the ideal triangle. A best triangle is a triangle through one interior angle equates to 90 degrees. In a right triangle, the two short legs accomplish at an edge of 90 degrees. The hypotenuse that a triangle is opposite the 90-degree angle.

What is the Pythagorean Theorem?

The Pythagoras theorem is a mathematical regulation that says that the sum of squares of the lengths that the two brief sides that the appropriate triangle is equal to the square that the size of the hypotenuse.

The Pythagoras to organize is algebraically written as:

a2 + b2 = c2

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They are attracted in such a method that they kind a best triangle. We deserve to write their locations can in equation form:

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

Let’s intend the size of square I, square II, and square III space a, b and also c, respectively.

Then,

Area of Square I = a 2

Area of Square II = b 2

Area that Square III = c 2

Hence, we have the right to write the as:

a 2 + b 2 = c 2

which is a Pythagorean Theorem.

 

The Converse of the Pythagorean Theorem

The converse the the Pythagorean theorem is a rule that is supplied to share triangles as either ideal triangle, acute triangle, or obtuse triangle.

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

For an acute triangle, c22 + b2, wherein c is the side opposite the acute angle.For a appropriate triangle, c2= a2 + b2, wherein c is the next 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 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, room 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 next 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 to organize formula is given as:

⇒ c2 = a2 + b2

where;

c = size of the hypotenuse;

a = size of one side;

b = size of the 2nd side.

We can use this formula to settle various troubles involving right-angled triangles. Because that instance, we deserve to use the formula to recognize the 3rd length that a triangle as soon as the lengths of two sides that the triangle are known.

Application of Pythagoras to organize formula in genuine Life

We can use the Pythagoras theorem to examine whether a triangle is a ideal triangle or not.In oceanography, the formula is supplied to calculate the rate of sound tide in water.Pythagoras organize is supplied in meteorology and also aerospace to identify the sound resource and the range.We deserve to use the Pythagoras to organize to calculate electronic contents such as tv screens, computer screens, solar panels, etc.We have the right to use the Pythagorean organize to calculation the gradient that a certain landscape.In navigation, the organize is offered to calculate the shortest distance between given points.In architecture and construction, we can use the Pythagorean to organize to calculation the slope of a roof, drainage system, dam, etc.

Worked instances of Pythagoras theorem:

Example 4

The two short sides the a appropriate triangle are 5 cm and 12cm. Discover the length of the third side

Solution

Given, a = 5 cm

b = 12 cm

c = ?

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

c2 = a2 + b2

c2 =122 + 52

c2 = 144 + 25

√c2 = √169

c = 13.

Therefore, the 3rd is same to 13 cm.

Example 5

The diagonal and one side length of a triangular next is 25cm and 24cm, respectively. What is the dimension 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 size of a computer screen who dimensions space 8 inches and also 14 inches.

Hint: The diagonal of the display is that size.

Solution

The size of a computer screen is the very 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 screen is 17 inches.

Example 7

Find the right triangle area offered that the diagonal and also the bases space 8.5 cm and 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 ideal triangle = (½) x basic x height

= (½ x 7.7 x 4) cm2

= 15.4 cm2

Practice Questions

A 20 m long rope is extended from the peak of a 12 m tree to the ground. What is the distance between the tree and also the finish of the rope on the ground?A 13 m long ladder is leaning against the wall. If the soil distance between the foot the the ladder and the wall is 5 m, what is the wall’s height?