Let O be an interior point of triangle ABC such that OA +2 OB +3 OC =0. Then the ratio of the area of triangle ABC to the area of triangle AOC is

Question

Let O be an interior point of triangle ABC such that OA +2 OB +3 OC =0. Then the ratio of the area of triangle ABC to the area of triangle AOC is (A) 2 (B) (3/2) (C) 3 (D) (5/3)

Tardigrade Admin · Accepted Answer

In the diagram, let D and E be the midpoints of the sides AC and BC, respectively. Then we have

O A + OC = 2 O D

.....(1)
and

2 (OB + OC) = 4 OE

....(2)
From equations (1) and (2), we get

O A + 2 OB + 3 OC = 2 (O D + 2 OE) = 0

It follows that

O D

and

OE

are collinear, and

∣ O D ∣ = 2∣ OE ∣

.
Consequently,

\frac{S _{△ A EC}}{S _{△ A OC}} = \frac{3}{2}

and

\frac{S _{△ A EC}}{S _{△ A OC}} = \frac{3 \times 2}{2} = 3

Q. Let $O$ be an interior point of $△ A BC$ such that $O A + 2 OB + 3 OC = 0$ . Then the ratio of the area of $△ A BC$ to the area of $△ A OC$ is

2

$\frac{3}{2}$

3

$\frac{5}{3}$

Q. Let O be an interior point of △ABC such that OA+2OB+3OC=0. Then the ratio of the area of △ABC to the area of △AOC is

2

23​

3

35​

Q. Let $O$ be an interior point of $△ A BC$ such that $O A + 2 OB + 3 OC = 0$ . Then the ratio of the area of $△ A BC$ to the area of $△ A OC$ is

$\frac{3}{2}$

$\frac{5}{3}$