Question

In $\Delta$ ABC, D and E are the midpoints of side AB and AC. State which of the following are correct option(s).

• A. Area of $\Delta$ BEC and $\Delta$ BDC are equal.
• B. Area of $\Delta$ BDE and $\Delta$ BDC are equal.
• C. Area of $\Delta$ BEC and $\Delta$ CED are equal.
• D. Area of $\Delta$ DOB and $\Delta$ EOC are equal.

Answer 1

Answer: (A), (D)

The correct options are
A Area of $\Delta$ BEC and $\Delta$ BDC are equal.
D Area of $\Delta$ DOB and $\Delta$ EOC are equal.

(i) D is midpoint of AB (given)

(ii) E is midpoint of AC (given)
From midpoint theorem DE is parallel to BC.

(iii) Area of $\Delta$ BEC and $\Delta$ BDC are equal.
($\Delta$'s on the same base(BC) and between the same parallels(DE, BC))

(iv) area $\Delta BDE$ = area $\Delta CED$
($\Delta$'s on the same base(DE) and between the same parallels(DE, BC))

$\therefore$ area $\Delta DOB$ + area $\Delta DOE=\text{area}\Delta DOE$ + area $\Delta EOC$

$\therefore$ area $\Delta DOB$ = area $\Delta EOC$