Pythagoras Theorem (also known as Pythagorean Theorem) is a mathematical concept that describes how the sides of a right-angled triangle are related.
Pythagoras theorem tells that “The square of the hypotenuse side is equal to the sum of squares of the other two sides, in a right-angled triangle”. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. The longest side of this triangle is hypotenuse to the opposite to 90^{o }.
Pythagorean triples are another name for the sides of a right triangle. This theorem's formula and proof are presented with examples.
The Pythagoras theorem is a formula for calculating the length of an unknown side and the angle of a triangle. We can derive the base, perpendicular, and hypotenuse formulas using this theorem. Let's take a closer look at the Pythagorean theorem's mathematics.
History of Pythagoras' Theorem-Right Angle Triangle:
The theorem is named after Pythagoras, a Greek mathematician.
Formula for Pythagoras' Theorem
Consider the following triangle:
The perpendicular is denoted by "a."
The base is "b."
The hypotenuse is represented by the letter "c."
The Pythagoras Theorem formula, according to the definition, is as follows:
Perpendicular^{2} + Base^{2} = Hypotenuse^{2}
a^{2} + b^{2} = c^{2}
Because the side opposing the largest angle is the longest (known as Hypotenuse), the side opposite the right angle (90°) is the longest.
Theorem of Pythagoras
Consider three squares with sides a, b, and c that are attached to the three sides of a triangle with the same sides as indicated.
Theorem of Pythagoras –
The area of squares "a" and "b" equals the area of square "c."
Proof of Pythagoras' Theorem: A right-angled triangle ABC with a right angle at B.
To Demonstrate: AC2 = AB2 + BC2
Draw a perpendicular BD that intersects AC at D.
Proof of Pythagoras' Theorem
Proof: We all know what ADB and ABC stand for.
Therefore,
(Similar triangles' equivalent sides)
Alternatively, AB2 = AD + AC ……………………………...……. (1)
Moreover, BDC ABC
Therefore,
(Similar triangles' equivalent sides)
Alternatively, BC2 = CD AC ……………………………………. (2)
When we combine equations (1) and (2), we get
AB2 + BC2 = AD AB2 + BC2 = AD AB2 + BC2 = AD AB2 + BC2 = AD AB2 + BC2 =
AC (AD + CD) = AB2 + BC2
Because AD + CD Equals AC
As a result, AC2 = AB2 + BC2.
As a result, the Pythagorean theorem is established.
The Pythagorean theorem applies solely to right-angled triangles.
Pythagoras Theorem Applications
To determine whether or not the triangle is a right-angled triangle.
If the other two sides of a right-angled triangle are known, we can compute the length of any side.
To determine a square's diagonal.
Useful Pythagoras' theorem can be used to determine the sides of a right-angled triangle. We can find the third side of a right triangle if we know the two sides.
Remember the following formula to apply Pythagoras' theorem:
a^{2} + b^{2} = c^{2}
If we have the lengths of three sides of a triangle, we may apply the Pythagorean theorem to determine if the triangle is a right-angled triangle or not.
FAQs – Frequently Asked Questions
In the case of a right-angled triangle, Pythagoras' formula is c2=a2+b2.
The hypotenuse is the longest side of a right-angled triangle, opposite the right angle and perpendicular to the base. Let a, b, and c stand for the base, perpendicular, and hypotenuse, respectively. The hypotenuse formula will be c^{2} = (a^{2} + b^{2}), as stated by Pythagoras.
Is the Pythagoras Theorem applicable to any triangle?
No, only the right-angled triangle is covered by this theorem.
The theorem can be used to calculate the steepness of mountains or hills. To calculate the distance between an observer and a location on the ground from a tower or building above which the observer is looking. It is mostly utilized in the building industry.