Write the correct answer of the following: ABCD is a trapezium wi… - Vidyadip

MCQ

Write the correct answer of the following: $ABCD$ is a trapezium with parallel sides $AB = a\ cm$ and $DC = b\ cm$ (Fig). $E$ and $F$ are the mid-points of the non-parallel sides. The ratio of ar $(ABFE)$ and ar $(EFCD)$ is:

A
$a : b$
✓
$(3a + b) : (a + 3b)$
C
$(a + 3b) : (3a + b)$
D
$(2a + b) : (3a + b)$

✓

Answer

Correct option: B.

$(3a + b) : (a + 3b)$

$ABCD$ is a trapizium in which $AB || DC$. $E$ and $F$ are the mid - point of $AD$ and $BC$, so
$\text{EF}=\frac{1}{2}(\text{a}+\text{b})$
$ABEF$ and $EFCD$ are also trapeziums.
$\text{ar}(\text{ABEF})=\frac{1}{2}\Big[\frac{1}{2}(\text{a}+\text{b})+\text{a}\Big]\times\text{h}=\frac{\text{h}}{4}(3\text{a}+\text{b})$
$\text{ar}(\text{EFCD})=\frac{1}{2}\Big[\text{b}+\frac{1}{2}(\text{a}+\text{b})\Big]\times\text{h}=\frac{\text{h}}{4}(\text{a}+3\text{b})$
$\therefore\frac{\text{ar}(\text{ABEF})}{\text{ar}(\text{EFCD})}=\frac{\frac{\text{h}}{4}(3\text{a}+\text{b})}{\frac{\text{h}}{4}(\text{a}+3\text{b})}=\frac{(3\text{a}+\text{b})}{(\text{a}+3\text{b})}$
So, the required ratio is $(3a + b) : (a + 3b)$.

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