Question

Prove that the area of a trapezium is
half the sum of the parallel sides mul-
tiplied by the distance between them.
​

Answer 1

Step-by-step explanation:

In figure, for triangle △ABC, we can see that height of △ABC is h and base is AB. and, for triangle △ADC, we can see that height of △ADC is h and base is CD. Hence, we proved that the area of trapezium is half of thesum of parallel sides multiplied by the distance between them.

Answer 2

Step-by-step explanation:

Complete step-by-step answer:

Before we solve the question, let us see what is trapezium and what the properties of trapezium are.

A trapezium is a 2-D shape and a type of quadrilateral, which has only two parallel sides and the other two sides are non-parallel, where quadrilateral means a polygon or closed shape with four sides and four vertices.

In trapezium, only one pair of opposite sides are parallel and none of sides are equal. A square, rectangle and rhombus are trapezium but trapezium is not a square, rectangle and rhombus.

Now, I question whether we have to prove that the area of trapezium is half of the sum of parallel sides multiplied by the distance between them.

Let, we have a trapezium ABCD, with a pair of parallel sides AB and DC and non – parallel sides AD and BC and height of trapezium is h.

So, we have to prove that Area of Trapezium ABCD =12×h×(AB+CD)

Now, firstly we will do some construction. We will firstly construct altitude from vertex A to side DC, and diagonal AC from vertex A to vertex C. Also, we will extend sides AB and a perpendicular line from vertex C meeting extended line AB at point O making angle of 90∘ .

Now from the figure we can see trapezium ABCD is divided into two triangles △ABC and △ADC.

So, area of trapezium = area of triangle △ABC + area of triangle △ADC ….. ( i )

Now, from the figure we can see, side AQ is equal to OC as both are perpendicular distance equals to height of trapezium h.

Now, we know that area of triangle whose height is h and base is d is equals to

12×h×d

In figure, for triangle △ABC, we can see that height of △ABC is h and base is AB.

So, area of triangle △ABC=12×h×AB

and, for triangle △ADC, we can see that height of △ADC is h and base is CD.

So, area of triangle △ADC=12×h×CD

Form statement ( i ), we can say that

Area of trapezium ABCD =12×h×AB+12×h×CD

Taking,

12×h, common, we get

Area of trapezium ABCD =12×h×(AB+CD)

Hence, we proved that the area of trapezium is half of the sum of parallel sides multiplied by the distance between them.