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Areas of Parallelograms and Triangles — MATHS STD 9 — Question

Rajasthan BoardEnglish MediumSTD 9MATHSAreas of Parallelograms and Triangles1 Mark
Question
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:
  1. a : b
  2. (3a + b) : (a + 3b)
  3. (a + 3b) : (3a + b)
  4. (2a + b) : (3a + b)
✓

Answer

  1. (3a + b) : (a + 3b)
Solution:
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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