Triangles as Parts of Rectangles

Triangles as Parts of Rectangles: Suppose we have a paper that is shaped like a rectangle. Fold the paper diagonally, such that the paper is divided into two parts. You can see the rectangle shaped paper is divided into two triangles.

Therefore, these triangles formed by folding the paper, are the parts of a rectangle. Now, if we find the area of these two triangles and add them together, the resulting value will be the area of the whole paper, i.e., the rectangle. Also, we can say, the two triangles are congruent to each other by SSS (Side, Side, Side) congruency.

Solved Example

Let us take an example of a rectangle having length = 8cm and width = 5cm.

So, Area of rectangle = Length x width

= 8cm x 5 cm

= 40 cm² ……………(i)

Now divide the rectangle, along its diagonal, into two triangles. The two triangles obtained are both right-angled triangles.

Height of the right triangles = 5cm each

Base of the right triangle = 8cm each

Now, by the formula of area of triangle we know;

Area of Triangle-1 = ½ (base) x (height)

= ½ (8cm) x (5cm)

= 20cm²

The area of another triangle will also be the same, since height and base values are the same. Therefore,

Area of Triangle-2 = 20cm²

Now if we add, the areas of the two triangles, we get;

Area of (Triangle-1 and Triangle-2) = 20cm² + 20cm²

Area = 40 sq.cm ………..(ii)

By equation (i) and (ii) we can say that;

By this we can also conclude that;

Triangles as parts of Square

Here also, the same theory is applicable to what we have understood for triangles as parts of rectangles.

A square has all its sides equal. Suppose we have a square with sides equal to ‘a’. Then by the formula of area of square;

Area (Square) = a x a = a² ………..(i)

Now, divide the square into two parts by cutting it diagonally. Again we got two triangles that superimposed on each other.

Since, the square has all its sides equal, therefore, on dividing it diagonally, we will get two right-isosceles triangles, having base and height equal to ‘a’.

Base = a unit

Height = a unit

Area of Triangle-1 = ½ a x a = a²/2

Area of Triangle-2 = ½ a x a = a²/2

By adding the areas of two triangles, we get;

Sum of areas of two triangles = a²/2 + a²/2 = 2(a²/2) = a² ……………(ii)

Therefore, by equation (i) and (ii), we get;

We can also conclude that:

In geometry, we have learned different shapes and sizes that are used to define the shapes in the real world. By learning the relationship between two or more shapes, we can easily determine the required dimensions or area.

Video Lesson on Triangles and its Properties