Hypotenuse Formula Proof
As Pythagorean theorem shows the relation in the sides of a right-angled triangle, so if the length of any side is missing, it can be calculated using the Pythagorean Theorem.
In a right triangle ABC, right-angled at B.
To Prove: Hypotenuse2 = Base2 + Perpendicular2
In the below triangle ABC, let us draw a line from B to touch the side AC at angle D.
By similar triangles theorem, we can write the following equations such that,
△ADB ~ △ABC
So we can write equation as
⇒ AD/AB = AB/AC
⇒ AB2= AD × AC . . .(i)
Again, △BDC ~△ABC
So, we can write equation as follows
⇒ CD/BC = BC/AC
⇒ BC2 = CD × AC . . .(ii)
Now, if we add (i) and (ii) we get;
AB2 + BC2 = (AD × AC) + (CD × AC)
Taking AC as common term from right side, we get;
⇒ AB2 + BC2 = AC × (AD + CD)
⇒ AB2 + BC2 = AC × (AC)
⇒ AB2 + BC2 = AC2
Hypotenuse2 = Base2+ Perpendicular2
Hypotenuse = √ (Perpendicular²+ Base²)
Note: By using the lengths of both a (Perpendicular) and b (Base) , then the length of c (Hypotenuse) can be calculated:
The formula to calculate Hypotenuse:
Hypotenuse =√(Perpendicular²+ Base²)
OR
c = √(a² + b²)
Hypotenuse Formula
Hypotenuse Formula or Hypotenuse Theorem Formula is another name for Pythagoras Theorem. Hypotenuse Formula is used to calculate the third side of the right-angled triangle given the other two sides. Hypotenuse Formula can be defined as a relation among the three sides (hypotenuse, base, perpendicular) of a right-angled triangle. Hypotenuse Formula states that the sum of squares of two small sides(base and perpendicular) is equal to the square of the longest side (hypotenuse).
In this article, we will explore all the basic details of the Hypotenuse Formula, including the mathematical expression, proof, and various solved examples using the Hypotenuse Formula.
Table of Content
- What is the Hypotenuse?
- Hypotenuse Formula in Triangle
- Hypotenuse Formula Proof
- Application of Hypotenuse Formula
- Sample Problems on Hypotenuse Formula