Question

What is the area of the plot shown in the figure?

A) $\dfrac{1}{2}\left( {az + by + ct + dx} \right)$
B) $\dfrac{1}{2}\left( {bt + cx + ay + az} \right)$
C) $\dfrac{1}{2}\left( {cx + bt + by + az} \right)$
D) $\dfrac{1}{2}\left( {d + t} \right)\left( {c + x} \right) + \dfrac{1}{2}\left( {a + b} \right)\left( {y + z} \right)$

Answer 1

Hint: The given figure consists of four triangles. The formula to find the area of the triangle when its base and height is given is: $\dfrac{1}{2}bh$ where $b$ stands for base and $h$ stands for height.

Complete step-by-step solution:
The total area of figure $ABCDEF$ is given as the sum of the area of the $\Delta ABC$,$\Delta BCD$,$\Delta BED$and$\Delta EFD$,that is given as:
$Ar\left( {ABCDEF} \right) = Ar\left( {\Delta ABC} \right) + Ar\left( {\Delta BCD} \right) + Ar\left( {\Delta BED} \right) + Ar\left( {\Delta EFD} \right)$ ……(i)
In triangle $\Delta ABC$,
Base of the triangle is $a$.
Height of the triangle is $z$.
Hence, area of the triangle is given by,
$
\Delta ABC = \dfrac{1}{2}bh\\
 = \dfrac{1}{2} \times a \times z……(ii)
$
In triangle $\Delta BCD$,
Base of the triangle is $a$.
Height of the triangle is $y$.
Hence, area of the triangle is given by,
$
\Delta BCD = \dfrac{1}{2}bh\\
 = \dfrac{1}{2} \times a \times y……(iii)
$
In triangle $\Delta BED$,
Base of the triangle is$b$ .
Height of the triangle is$t$.
Hence, area of the triangle is given by,
$
\Delta BED = \dfrac{1}{2}bh\\
 = \dfrac{1}{2} \times b \times t……(iv)
$
In triangle $\Delta EFD$,
Base of the triangle is $c$.
Height of the triangle is $x$.
Hence, area of the triangle is given by,
$
\Delta EFD = \dfrac{1}{2}bh\\
 = \dfrac{1}{2} \times c \times x ……(v)
$
Substitute the values of areas of the triangles $\Delta ABC$,$\Delta BCD$,$\Delta BED$and$\Delta EFD$from equation (ii), (iii), (iv) and (v) respectively in equation (i).
$
Ar\left( {ABCDEF} \right) = \dfrac{1}{2}az + \dfrac{1}{2}ay + \dfrac{1}{2}bt + \dfrac{1}{2}cx\\
 = \dfrac{1}{2}\left( {az + ay + bt + cx} \right)
$

Therefore, option (B) is the correct answer.

Note: In such types of problems, make sure to find the correct value of base and height for the corresponding triangles to find the total area of the figure. The solution is totally based on the formula of area of the triangle.