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.

Triangle

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:

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

Right Angled 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.

Right Angle Triangle

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.

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.

Right Angle Triangle

Where, s is the semiperimeter 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.

Right Angled Triangle

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:

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

Right Angled Triangle

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

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