# Right Angled Triangle

## What is Triangle?

A triangle is a regular polygon, with three sides and the sum of any two sides is always greater than the third side. This is a unique property of a triangle. In other definition, it can be said as any closed figure with three sides with its sum of angles equal to 180.

Being a closed figure, a triangle can have different shapes, and each shape is described by the angle made by any two adjacent sides.

## Types of Triangles:

**Acute angle triangle:**When the angle between 2 sides is less than 90 it is called an acute angle triangle.**Right angle triangle:**When the angle between any two sides is equal to 90 it is called as right angle triangle.**Obtuse angle triangle:**When the angle between any two sides is greater than 90 it is called an obtuse angle triangle.

## Right Angled Triangle

A Right-angled triangle is one of the most important shapes in geometry and is the basics of trigonometry. A right-angled triangle is the one which has 3 sides, “base” “hypotenuse” and “height” with the angle between base and height being 90°. But the question arises what are these? Well, these are the three sides of a right-angled triangle and generates the most important theorem that is Pythagoras theorem.

The area of the biggest square is equal to the sum of the square of the two other small square area. We can generate Pythagoras as the square of the length of the hypotenuse is equal to the sum of the length of squares of base and height

We can generate Pythagoras as the square of the length of the hypotenuse is equal to the sum of the length of squares of base and height

As now we have a general idea about the shape and basic property of a right-angled triangle, let us discuss the area of a triangle.

## Right Angle Triangle Properties

Let us discuss, the properties carried by a right angle triangle.

- One angle is always 90° or right angle.
- The side opposite angle 90° is the hypotenuse.
- The hypotenuse is always the longest side.
- The sum of the other two interior angles is equal to 90°.
- The other two sides adjacent to the right angle are called base and perpendicular.
- The area of right angle triangle is equal to half of the product of adjacent sides of the right angle, i.e.,

**Area of Right Angle Triangle = ½ (Base × Perpendicular)**

- If we drop a perpendicular from the right angle to the hypotenuse, we will get three similar triangles.
- If we draw a circumcircle which passes through all three vertices, then the radius of this circle is equal to half of the length of the hypotenuse.
- If one of the angles is 90° and the other two angles are equal to 450 each, then the triangle is called an Isosceles Right Angled Triangle, where the adjacent sides to 90° are equal in length to each other.

Above were the general properties of Right angle triangle. The construction of the right angle triangle is also very easy. Keep learning with CT’S to get more such study materials related to different topics of Geometry and other subjective topics.

## Area of Right Angled Triangle

The area is in 2 dimensional and is measured in square unit.it can be defined as the amount of space taken by the 2-dimensional object.

The area of a triangle can be calculated by 2 formulas:

area= \(\frac{a \times b }{2}\)

and

Heron’s formula i.e. area= \(\sqrt{s(s-a)(s-b)(s-c)}\),

where s =, and a,b,c are the sides of a triangle.

Where, s is the semi perimeter and is calculated as s \(=\frac{a+b+c}{2}\) and a, b, c are the sides of a triangle.

Let us calculate the area of a triangle using the figure given below.

**Fig 1:** Let us drop a perpendicular to the base b in the given right angle triangle Now let us multiply the triangle into 2 triangles.

**Fig 2:** It forms a shape of a parallelogram as shown in the figure.

**Fig 3:** Let us move the yellow shaded region to the beige colored region as shown the figure.

**Fig 4:** It takes up the shape of a rectangle now.

Now by the property of area, it is calculated as the multiplication of any two sides

Hence area =b×h (for a rectangle)

Therefore, the area of a right angle triangle will be half i.e.

= \(\frac{b \times h}{2}\)

For a right angled triangle, the base is always perpendicular to the height. When the sides of the triangle are not given and only angles are given, the area of a right-angled triangle can be calculated by the given formula:

= \(\frac{bc \times ba}{2}\)

Where a, b, c are respective angles of the right angle triangle, with ∠b always being 90°.

## Frequently Asked Questions From Right Angle Triangle

### What are Right Angled Triangles?

Right-angled triangles are those triangles in which one angle is 90 degrees. Since 1 angle is 90°, the sum of the other two angles will be 90°.

### How to Find the Missing Side of any Right Angled Triangle?

For a right-angled triangle, trigonometric functions or the Pythagoras theorem can be used to find its missing sides. If two sides are given, the Pythagoras theorem can be used and when the measurement of 1 side and an angle is given, trigonometric functions like sine, cos, and tan can be used.

### Can a Triangle have Two Right Angles? Explain.

No, a triangle can never have 2 right angles. A triangle has exactly 3 sides and the sum of interior angles sum up to 180°. So, if a triangle has two right angles, the third angle will have to be 0 degrees which means the third side will overlap with the other side. Thus, it is not possible to have a triangle with 2 right angles.