Hypotenuse

In Mathematics, the term “Hypotenuse” comes from the Greek word hypoteinousa that means “stretching under”. This term is used in Geometry, especially in the right angle triangle. The longest side of a right-angle triangle is called the hypotenuse (the side which is opposite to the right angle). In this article, we are going to discuss the meaning of the term hypotenuse along with its formula, theorem, proof and examples.

Hypotenuse Meaning

Hypotenuse means, the longest side of a right-angled triangle compared to the length of the base and perpendicular. The hypotenuse side is opposite to the right angle, which is the biggest angle of all the three angles in a right triangle. Basically, the hypotenuse is the property of only the right triangle and no other triangle. Now, this is better explained when we learn about the right-angled theorem or Pythagoras Theorem or Pythagorean theorem. These concepts are majorly used in Trigonometry. Here, in this article, we will learn in detail about hypotenuse, its formula, theorem along with examples.

Hypotenuse Theorem

The hypotenuse theorem is defined by Pythagoras theorem, According to this theorem, the square of the hypotenuse side of a right-angled triangle is equal to the sum of squares of base and perpendicular of the same triangle, such that;

Hypotenuse2 = Base2 + Perpendicular2

Hypotenuse Formula

The formula to find the hypotenuse is given by the square root of the sum of squares of base and perpendicular of a right-angled triangle. The hypotenuse formula can be expressed as;

Hypotenuse = √[Base2 + Perpendicular2]

Let a, b and c be the sides of the triangle as per given figure below;

Hypotenuse Formula

So the hypotenuse formula for this triangle can be given as;

c2 = a2 + b2

Where a is the perpendicular, b is the base and c is the hypotenuse.

Hypotenuse Theorem Proof

Given: A right triangle ABC, right-angled at B.

To Prove: Hypotenuse2 = Base2 + Perpendicular2

Hypotenuse theorem proof

Proof: In triangle ABC, let us draw a line from B to touch the side AC at D.

By similar triangles theorem, we can write;

△ADB ~ △ABC

So, AD/AB = AB/AC

Or AB2 = AD x AC ………………..1

Again, △BDC ~△ABC

So, CD/BC = BC/AC

Or

BC2 = CD x AC ……………2

Now, if we add eq. 1 and 2 we get;

AB2 + BC2 = (AD x AC) + (CD x AC)

Taking AC as common term from right side, we get;

AB2 + BC2 = AC (AD + CD)

AB2 + BC2 = AC (AC)

AB2 + BC2 = AC2

Base2 + Perpendicular2 = Hypotenuse2

Hence, proved.

Hypotenuse of a triangle

The hypotenuse is only defined for the right-angled triangle. It is not defined for any other types of triangles in geometry such as

  • Acute Angled Triangle
  • Obtuse Angled Triangle
  • Scalene Triangle
  • Isosceles Triangle
  • Equilateral Triangle

But only isosceles triangle could be represented as a right-angled triangle, where the length of the base side and perpendicular side are equal and the third side will be the hypotenuse.

Hypotenuse Examples

Let us solve some examples based on the hypotenuse concept.

Example 1:

If the base and perpendicular of a right-angled triangle are 3cm and 4cm, respectively, find the hypotenuse.

Solution:

Given, base = 3cm and perpendicular = 4cm

By the hypotenuse formula, we know;

Hypotenuse = √(Base2 + Perpendicular2)

= √(32 + 42)

= √(9 + 16)

= √25

= 5cm

Hence, the length of the hypotenuse is 5cm.

Example 2:

For an isosceles right-angled triangle, the two smallest sides are equal to 10cm. Find the length of the longest side.

Solution:

The two equal sides of the isosceles right triangle are the base and perpendicular.

The longest side is the hypotenuse. Hence, by using the formula, we get;

Hypotenuse = √(Base2 + Perpendicular2)

H = √(102 + 102)

H = √(100 + 100)

H = √10000

H = 100

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