AREA OF 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 words, it can be said that any closed figure with three sides and the sum of all the three internal 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:

1. Acute angle triangle: When the angle between any 2 sides is less than 90 degrees it is called an acute angle triangle.
2. Right angle triangle: When the angle between a pair of sides is equal to 90 degrees it is called a right-angle triangle.
3. Obtuse angle triangle: When the angle between a pair of sides is greater than 90 degrees 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..

Properties of 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 the 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.

Area of Right Angled Triangle

The area is in the two-dimensional region and is measured in a 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= a×b2 and

Heron’s formula i.e. area= s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√,

Area of a Right-Angled Triangle

Draw a diagonal and cut out the rectangle.  Then cut along the diagonal to form two right-angled triangles.

By arranging one triangle over the other, we find that the triangles are congruent.  In other words, the triangles are the same size and thus, equal in area.  This suggests that the area of a triangle is equal to half the area of a rectangle around it.  Therefore:

In the diagram, we notice that the length of the rectangle is one side of the triangle.  This is said to be the base of the triangle.  So:

Base of the triangle = Length of the rectangle

The distance from the top of the triangle to the base is called the height of the triangle.  Therefore:

Height of the triangle = Width of the rectangle

Replacing l and w with the Base and Height in equation (1), we obtain:

Using the pronumerals A for area, b for base and h for height, we can write the formula for the area of a right-angled triangle as:

Area of a Triangle

Consider the following triangle.

Enclose the triangle by drawing a rectangle around it as shown below.

It is clear from the diagram that the length of the rectangle is one side of the triangle.  This is said to be the base of the triangle.  So:

Base of the triangle = Length of the rectangle

The distance from the top of the triangle to the base is called the height of the triangle.  Clearly:

Height of the triangle = Width of the rectangle

Using the pronumerals A for area, b for base and h for height, we can write the formula for the area of a triangle as:

Note:

The rule (or equation)

represents the relationship between the nase and height of a triangle and its area.  Such an equation, which gives a rule for working out the value of one quantity from the values of others is called a formula.

Just to recap the ongoing discussion:

Find the area of a triangle with base 8 cm and height 5 cm.

Solution