AP EAMCET · Maths · Vector Algebra

Let $D$ and $E$ be the midpoints of the sides $A C$ and $B C$ of a triangle $A B C$ respectively. If $O$ is an interior point of the triangle $A B C$ such that $\vec{O A} + 2 \vec{O B} + 3 \vec{O C} = \vec{0},$ then the area (in sq. units) of the triangle $O D E$ is

A $6$
B $5$
C $\frac{3}{4}$
D $0$

Verified Solution

Answer & Solution

Correct Answer

(D) $0$

Step-by-step Solution

Detailed explanation

It is given that, OA→+2OB→+3OC→=0 The figure below represents the triangle ABC and midpoints. Ar△ODE=12|OD→×OE→| =12a→+c→2×b→+c→2…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let D and E be the midpoints of the sides AC and BC of a triangle ABC respectively. If O is an interior point of the triangle ABC such that OA→+2OB→+3OC→=0→, then the area (in sq. units) of the triangle ODE is

Step-by-step Solution

Let $D$ and $E$ be the midpoints of the sides $A C$ and $B C$ of a triangle $A B C$ respectively. If $O$ is an interior point of the triangle $A B C$ such that $\vec{O A} + 2 \vec{O B} + 3 \vec{O C} = \vec{0},$ then the area (in sq. units) of the triangle $O D E$ is