Hypotenuse

The name “hypotenuse” is derived from the Greek word hypotenuse, which means “extending under.” This phrase is used in Geometry, particularly in relation to the right angle triangle. The hypotenuse refers to the longest side of a right-angle triangle (the side which is opposite to the right angle). The definition of the term hypotenuse, as well as its formula, theorem, proof, and examples, will be discussed in this article.

Hypotenuse

The hypotenuse is the longest side of a right-angled triangle when compared to the base and perpendicular lengths. The hypotenuse side of a right triangle is opposite the right angle, which is the largest of the three angles. The hypotenuse is a quality that only the right triangle has, and no other triangles have. When we learn about the right-angled theorem, also known as Pythagoras Theorem or Pythagorean Theorem, this becomes clearer. Trigonometry makes extensive use of these notions.

Hypotenuse Theorem

Pythagoras’ theorem defines the hypotenuse theorem. According to this theorem, the square of a right-angled triangle’s hypotenuse side is equal to the sum of the squares of the base and perpendicular sides, resulting in Hypotenuse2 = Base2 + Perpendicular2.

Hypotenuse Formula

The square root of the sum of squares of the base and perpendicular of a right-angled triangle is used to calculate the hypotenuse. Hypotenuse = [Base2 + Perpendicular2] is the formula for the hypotenuse.

Assume that the triangle’s sides are a, b, and c, as shown in the diagram below.

Formula for Hypotenuse

As a result, the hypotenuse formula for this triangle is c2 = a2 + b2.

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

Hypotenuse Theorem Proof

Given: A right triangle XYZ, right-angled at Y.

To Prove: Hypotenuse2 = Base2 + Perpendicular2

Proof: In triangle XYZ, let us draw a line from Y to touch the side XYZ at O.

By similar triangles theorem, we can write;

△XOY ~ △ XYZ

So, XO/XY = XY/ XZ

Or XY2 = XO x XZ ………………..1

Again, △YOZ ~△ XYZ

So, ZO/YZ = YZ / XZ

Or

YZ2 = ZO x XZ ……………2

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

XY2 + YZ2 = (XO x XZ) + (ZO x XZ)

Taking XZ as a common term from the right side, we get;

XY2 + YZ2 = XZ (XO + ZO)

XY 2 + YZ2 = XZ (XZ)

XY 2 + YZ2 = XZ 2

Base2 + Perpendicular2 = Hypotenuse2

Hence, proved.

The hypotenuse of a triangle

Only the hypotenuse of a right-angled triangle is defined. Other sorts of triangles in geometry, such as Acute Angled Triangles, are not defined such as

• Acute Angled Triangle

• Obtuse Angled Triangle

• Scalene Triangle

• Isosceles Triangle

• Equilateral Triangle

Only the isosceles triangle may be represented as a right-angled triangle, with the base and perpendicular sides equal in length and the hypotenuse on the third side.

How to Find the Altitude on a Hypotenuse?

The line segment that links the vertex to the side opposite the vertex is known as the altitude of a triangle in mathematics. The “altitude” refers to the length of the altitude. Similarly, the height of the hypotenuse is the line that runs through the perpendicular from the hypotenuse of a right triangle to the vertex opposite the hypotenuse. The “Geometric Mean Theorem” or “Right Triangle Altitude Theorem” explains the correlation between the altitude on the hypotenuse of a right triangle and the line segment generated on the hypotenuse in elementary geometry. The hypotenuse’s altitude is calculated as follows:

Here 

a = perpendicular side = 3 cm

b= base side = 4 cm

C = hypotenuse = 5 cm

The smallest side of the right triangle is 3 cm in this case.

As a result, when the perpendicular side is divided by the hypotenuse, we get

Altitude = ⅗ = 0.6

Now, multiply the result by the base side of the right triangle.

Altitude = 0.6 x 4

Altitude = 2.4 cm 

As a result, the hypotenuse of a right triangle has a height of 2.4 cm.

Similarly, trigonometry can be used to determine altitude. The lesser triangles’ angles are the same as the main right triangle’s angles. We may calculate the altitude using trigonometric formulas.

Hypotenuse Illustrations

Let’s have a look at some hypotenuse-based instances.

1. 1st example:

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

Solution:

Given that the base is 3cm and the perpendicular is 4cm

Hypotenuse = (Base2 + Perpendicular2) = (32 + 42) = (9 + 16) = 25 = 5cm, according to the hypotenuse formula.

As a result, the hypotenuse is 5cm long.

1. Example 2: The two smallest sides of an isosceles right-angled triangle are both equal to 10cm. Find the longest side’s length.

Solution:

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

The hypotenuse is the longest side. As a result of applying the formula, we obtain:

(Base2 + Perpendicular2) Hypotenuse

(102 + 102) H = (102 + 102) H = (102 + 102) H = (102 + 10

(100 + 100) H = (100 + 100) H = (100 + 100) H = (100 + 100

H = 102 cm H = 200 cm H = 200 cm H = 200 cm H = 200 cm H = 200 cm H = 200 cm

Conclusion

• Hypotenuse Meaning

• Theorem

• Formula

• Proof

• The hypotenuse of a Triangle

• Altitude to a Hypotenuse

• Examples