If any triangle and rectangle are on the same base and between the same parallel then let us prove logically that the area of triangular region is half the area in the shape of rectangular.

Given.

If any triangle and rectangle are on the same base and between the same parallel.

Formula used.

Area of rectangle = Length × Breadth

Area of triangle = × Base × Height

⇒ Property of parallel lines

Perpendicular distance between 2 parallel is always same.

Draw a rectangle ABCD between 2 parallel lines PQ and RS

Draw triangle with base CD and point E on common line AB

As we know that rectangle possess each angle to be 90°

So Breadth AD and BC act as perpendicular between parallel lines

Hence, Height of triangle(EF) = Breadth of rectangle(AD)

⇒ By Property of parallel lines

As both rectangle and triangle possess same base CD

Which is also length of rectangle

∴ Length of rectangle = Base of triangle = CD

Area of triangle = × Base × Height

× CD × EF

× CD × AD ∵ EF = AD

× [Length of rectangle × Breadth of rectangle]

× [Area of rectangle]

Hence proved;