Name: Figures on the Same Base
Uploaded: 2019-07-19
Duration: 6 min 34 s
Description: Two figures are said to be on one base and between the same parallels if they have a common side as the base and certain vertices lie on the line parallel to...

Question

Prove that the area of a trapezium is half the sum of the parallel sides multiplied by the distance between them.

Accepted Answer

$A B C D$ be trapezium with $C D | | A B$ .

$C F$ and $D H$ are perpendiculars to segment $A B$ from $C$ and $D$ respectively.

From figure,

Area of trapezium $A B C D =$ area $(△ A F C) +$ area of rectangle $C D F H +$ area $(△ B H D) \dots (i)$

Consider rectangle $C D H F$ ,

Length $= F H$

Breadth $= C F$

Area of rectangle $=$ length $\times$ breadth

The area of rectangle $C D F H = F H \times C F . . . (ii)$

Consider $△ A F C$ ,

Base $= A F$

Height $= C F$

$\Rightarrow$ area $(△ A F C) = \frac{1}{2} \times A F \times C F . . . . (iii)$

Consider $△ D B H$ ,

Base $= B H$

Height $= H D$

$\Rightarrow$ area $(△ D B H) = \frac{1}{2} \times B H \times H D . . . . (iv)$

Substitute $(ii)$ , $(iii)$ and $(iv)$ in $(i)$ we get,

Area of trapezium $A B C D = F H \times C F + \frac{1}{2} \times A F \times C F + \frac{1}{2} \times B H \times H D$

Since $C D H F$ is rectangle,

$C F = H D = h$

$\Rightarrow$ Area of trapezium $A B C D = F H \times h + \frac{1}{2} \times A F \times h + \frac{1}{2} \times B H \times h$

$\Rightarrow$ Area of trapezium $A B C D = h \times (F H + \frac{1}{2} \times A F + \frac{1}{2} \times B H)$

$= h \times [F H + \frac{1}{2} \times (A F + B H)]$

$= h \times [F H + \frac{1}{2} \times (A B - F H)]$

$= h \times (F H + \frac{1}{2} \times A B - \frac{1}{2} \times F H)$

$= h \times (\frac{1}{2} \times F H + \frac{1}{2} \times A B)$

$= \frac{1}{2} \times h \times (F H + A B)$

Since $C D H F$ is rectangle,

$F H = C D$

$\Rightarrow$ Area of trapezium $A B C D = \frac{1}{2} \times h \times (C D + A B)$

$h$ is the distance between parallel sides $A B$ and $C D$ .

Therefore, area of a trapezium is half the sum of the parallel sides multiplied by the distance between them.

Prove that the area of a trapezium is half the sum of the parallel sides multiplied by the distance between them.

Important Points to Remember in Chapter -1 - Areas from Telangana Board Mathematics Class 9 Solutions

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