In geometry, you come across different types of figures, the properties of which, set them apart from one another. One common figure among them is a triangle. A triangle is a closed figure, a polygon, with three sides. It has 3 vertices and its 3 sides enclose 3 interior angles of the triangle. The sum of the three interior angles in a triangle is always 180 degrees.

The most common types of triangle that we study about are equilateral, isosceles, scalene and right angled triangle. In this section, we will talk about the right angled triangle, also called right triangle, and the formulas associated with it.

A right triangle is the one in which the measure of any one of the interior angles is 90 degrees. It is to be noted here that since the sum of interior angles in a triangle is 180 degrees, only 1 of the 3 angles can be a right angle. If the other two angles are equal, that is 45 degrees each, the triangle is called an isosceles right angled triangle. However, if the other two angles are unequal, it is a scalene right angled triangle.

The most common application of right angled triangles can be found in trigonometry. In fact, the relation between its angles and sides forms the basis for trigonometry.

## Formulas

\(Area~ of~ a~ right~ triangle = \frac{1}{2} bh\)

Where b and h refer to the base and height of triangle respectively.

- \(Perimeter ~of ~a~ right ~triangle = a+b+c\)

Where a, b and c are the measure of its three sides.

- Pythagoras Theorem defines the relationship between the three sides of a right angled triangle. Thus, if the measure of two of the three sides of a right triangle is given, we can use the Pythagoras Theorem to find out the third side.

**\(Hypotenuse^{2} = Perpendicular^{2} + Base^{2}\)**

In the figure given above, ∆ABC is a right angled triangle which is right angled at B. The side opposite to the right angle, that is the longest side, is called the hypotenuse of the triangle. In ∆ABC, AC is the hypotenuse. Angles A and C are the acute angles. We name the other two sides (apart from the hypotenuse) as the ‘base’ or ‘perpendicular’ depending on which of the two angles we take as the basis for working with the triangle.

**Derivation:**

Consider a right angled triangle ABC which has B as 90 degrees and AC is the hypotenuse.

Now we flip the triangle over its hypotenuse such that a rectangle ABCD with width h and length b is formed.

You already know that area of a rectangle is given as the product of its length and width, that is, length x breadth.

Hence, area of the rectangle ABCD = b x h

As you can see, the area of the right angled triangle ABC is nothing but one-half of the area of the rectangle ABCD.

Thus, \(Area ~of \Delta ABC = \frac{1}{2} Area ~of~ rectangle ABCD\)

Hence, area of a right angled triangle, given its base b and height

\(A= \frac{1}{2} bh\)

**Solved Examples:**

**Question 1:** The length of two sides of a right angled triangle is 5 cm and 8 cm. Find:

- Length of its hypotenuse
- Perimeter of the triangle
- Area of the triangle

**Solution:** Given,

One side a = 5cm

Other side b = 8 cm

- The length of the hypotenuse is,

Using Pythagoras theorem,

\(Hypotenuse^{2} = Perpendicular^{2} + Base^{2}\)

\(c^{2} = a^{2} +b^{2}\)

\(c^{2} = 5^{2} +8^{2}\)

\(c= \sqrt{25+64}= \sqrt{89}= 9.43cm\)

Perimeter of the right triangle = a + b + c = 5 + 8 + 9.43 = 22.43 cm

\(Area ~of~ a~ right ~triangle = \frac{1}{2} bh\)

Here, area of the right triangle = \(\frac{1}{2} (8\times5)= 20cm^{2}\)

**Question 2:** The perimeter of a right angled triangle is 32 cm. Its height and hypotenuse measure 10 cm and 13cm respectively. Find its area.

**Solution:** Given,

Perimeter = 32 cm

Hypotenuse a= 13 cm

Height b= 10 cm

Third side, c=?

We know that perimeter = a+ b+ c

32 cm = 13 + 10 + c

Therefore, c = 32 – 23 = 9 cm

\(Area = \frac{1}{2} bh = \frac{1}{2} (9\times10)= 45cm^{2}\)

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