Area Of Right Triangle

Theorem: Area of a right triangle is equal to one half of the product of its sides connected by a right angle.

Prerequisites:
Parallelogram (definition)
Rectangle (definition)
Properties of parallelogram (proof)
Area of a rectangle (proof)

Proof:

Let ABC be a given right triangle. Let us draw a line through C parallel to AB and a line through A parallel to BC. Let these two lines meet at D. Thus, the resulting figure ABCD is a parallelogram (see definition of parallelogram).

By using the properties of parallelogram, i.e., adjacent angles are supplementary and opposite angles are equal, since $\angle B = 90^o$, therefore all the angles are equal to $90^o$. Hence ABCD is a rectangle (see definition of rectangle). Hence, its area = $ab$ (see area of rectangle).

Also since diagonal of a parallelogram divides it into two congruent triangles (see properties of parallelogram), therefore, $\triangle ABC\cong\triangle ADC$, and thus they have the same area.

$\therefore\;\quad\text{Area of }\triangle ABC + \text{Area of }\triangle ABC = ab$
$\Rightarrow\quad\text{Area of }\triangle ABC = \frac{1}{2} ab$

Q.E.D.


Recommended:
Area of triangle
Area of rhombus
Pythagoras Theorem