The Pythagorean theorem is sometimes referred to as Pythagoras theorem. It is a fundamental concept in Euclidean geometry. It provides the relation between three sides of a right-angled triangle.

The Pythagoras theorem says that the square of the hypotenuse is equal to the sum of the squares of the perpendicular and the base.

The Pythagorean Theorem formula is given as:

Where H is the length of the hypotenuse, P is the length of the perpendicular, and B is the length of the base.

In a right-angled triangle, the side opposite to the right angle is the largest and is called the hypotenuse. While the other two sides are called the base and perpendicular. The side opposite to the angle you’re interested in is called the perpendicular and the other one becomes the base.

According to the Pythagorean Theorem formula, if two of the sides are known the third can be calculated using the above-mentioned relation.

The Pythagorean Theorem is named after Pythagoras who was a Greek mathematician. Although evidence indicates that the ancient mathematicians of Babylon, Mesopotamia, China, and India might have used the theorem, Pythagoras is credited to be the first one to provide mathematical proof for it. The others also provided proofs but for some special cases of the theorem.

We now have several proofs of the theorem; in fact, Pythagorean Theorem has the most proofs than any other mathematical theorem. It has geometric as well as algebraic proofs dating thousands of years back.

Moreover, the theorem can be generalized to provide relations for higher-dimensional spaces and even non-Euclidean spaces. In fact, it can be generalized for objects that are not right-angled triangles to n-dimensional objects that are not even triangles.

The Pythagoras theorem is not only used by mathematicians, it is also used by businessmen and other people outside the mathematical world. The theorem is taken as the symbol of mathematical perplexity, mystique, and intellectual power. It is a popular reference in plays, songs, stamps, cartoons, and literature.

As discussed above, the Pythagoras theorem is thought to date back to even before Pythagoras. However, he is most probably the first one to prove it. This is called the proof by rearrangement.

By William B. Faulk - Own work, CC BY-SA 4.0

Observe the two larger squares shown in the figure. Each square has four triangles. The difference between them is that the triangles are arranged in a different manner. This means that the white portion of the larger squares must have the same area. If you equate the two areas, you will end up with the Pythagorean Theorem formula shown in figure.

Here, c is taken as the length of the hypotenuse, a is the length of the perpendicular, and b is the length of the base.

The area of the smaller square in the first figure is:

Area1=cxc=c2

The area of the smaller squares in the other figure is:

Area2=a2

and

Area3=b2

If both figures are same and only differ in the manner in which their triangles are arranged, then these areas can be equated as:

Area1=Area2+Area3

c2=a2+b2

This proof is called the Pythagorean proof; however, several other proofs are also available. This particular proof is also described in the writings of Proclus, another Greek philosopher of a later time.

As we discussed before that the Pythagorean Theorem Formula is given by:

a2+ b2= c2

Here, c is the length of the hypotenuse, a is the length of the perpendicular, and b is the length of the base.

Let’s suppose you know the length of the base and perpendicular, you can calculate the length of hypotenuse by using the formula:

c = a2+ b2

If you are given the length of the hypotenuse and one side, either a or b, the other side can be found out by using the following formula:

a = c2- b2

or

b = c2- a2

So, we can say that if any of the two sides of a right-angled triangle are given, the third side can be determined using the Pythagorean theorem formula. However, the length of the hypotenuse will always be greater than the other two sides and always remain less than their sum.

The Pythagoras theorem forms the base of several other theorems and formulas. One of these is the law of cosines.

Using the law of cosines, you can determine any side of a triangle if the other two sides and the angle between them is given. The law of cosines is given by the formula:

c2= a2+ b2- 2abcos γ

Here a, b, and c are the sides of the triangle. γ is the angle between a and b, and is opposite to the side c.

If the angle γ is equal to 90 degrees, then its cosine will be zero. The law of cosines is therefore reduced to the Pythagoras theorem formula.

c2= a2+ b2

The distance formula states that:

d=∆x2+∆y2

Here

x = x1 - x2 and y = y1 - y2

If you square on both sides, you end up with the Pythagoras Theorem:

d2 = x2 + y2

If a right triangle has a 2m base and 3m perpendicular, what is the length of the hypotenuse?

Solution:

Using the Pythagoras Theorem, we know that

c2=a2+b2

So,

c=a2+b2

The length of the hypotenuse then comes out to be,

c=22+32=13m

Find the length of the base of a right triangle if the length of the hypotenuse is 55m and that of the perpendicular is 10m.

Solution:

Using the Pythagoras Theorem, we know that

c2=a2+b2

Separating the length of the base, we get

b2=c2-a2

b=c2-a2

Using the values given in the question, the base is found to be

b=552-102=54.083m

Consider a right triangle abc. If the base is 30m wide and the hypotenuse is measured to be 50m, find out the length of the perpendicular of the triangle.

Solution:

Using the Pythagoras theorem, we know that

c2=a2+b2

Now, simple separate the length of the perpendicular to get

a2=c2-b2

And

a=c2-b2

Using the values given in the question, we find the perpendicular as,

a=502-302

a=900

a=30m

If the lengths of the sides of a triangle ABC are given below, determine whether it is a right-angled triangle or not.

AB=45, BC=55, CA=75

Solution:

For any right-angled triangle, the Pythagoras theorem should be satisfied. We know that in a right-angled triangle the largest of the three sides is the hypotenuse. Therefore, in this case, we will consider CA=75 as the hypotenuse. Furthermore, let AB=45 be the base and BC=55 be the perpendicular.

Now, the Pythagoras Theorem states that,

c2=a2+b2

Taking the left hand-side of the equation:

c2=752

c2=5625

Now, taking the right hand-side of the equation,

a2+b2=452+552

a2+b2=2025+3025

a2+b2=5050

Since, left hand-side and right hand-side of the equations are not equal; therefore, the given triangle is not a right-angled triangle.

Repeat example 4 with the triangle PQR. Where:

PQ=28, QR=53, and PR=45.

Solution:

For any right-angled triangle the Pythagoras theorem should be saitsfied. We know that in a right-angled triangle the largest of the three sides is the hypotenuse. Therefore, in this case, we will consider QR=53 as the hypotenuse. Furthermore, let PQ=28 be the base and PR=45 be the perpendicular.

Now, the Pythagoras Theorem states that,

c2=a2+b2

Taking the left hand-side of the equation:

c2=532

c2=2809

Now, taking the right hand-side of the equation,

a2+b2=282+452

a2+b2=784+2025

a2+b2=2809

Since, left hand-side and right hand-side of the equations are equal; therefore, the given triangle is a right-angled triangle.

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