Question

Question 4
Write True or False and justify your answer :
ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Then, ar ($\Delta$ BDE) = ar ($\Delta$ ABC).

Answer 1

False.
Given,
$\Delta$ ABC and $\Delta$ BDE are two equilateral triangles.
$\therefore$ Area of $\Delta ABC=\frac{\sqrt{3}}{4}\times (side{)}^2=\frac{\sqrt{3}}{4}(BC{)}^2$
[ $\because$ in equilateral $\Delta$ ABC, AB = BC = AC]
Given, D is the mid-point of BC.

$\therefore BD=DC=\frac{1}{2}BC$
Now, area of an equilateral $\Delta BDE=\frac{\sqrt{3}}{4}\times (side{)}^2$
$=\frac{\sqrt{3}}{4}\times (BD{)}^2$
[$\because$ in equilateral $\Delta$ BDE, BD = DE = BE]
$=\frac{\sqrt{3}}{4}\times {\left(\frac{1}{2}BC\right)}^2$
$=\frac{\sqrt{3}}{4}\times \frac{1}{4}B{C}^2=\frac{1}{4}\times \left(\frac{\sqrt{3}}{4}B{C}^2\right)$
Area of $\Delta$ BDE = $\frac{1}{4}$ area of $\Delta$ ABC