Question

ABC is a triangle. E and F are mid points of AC and AB respectively. If the area of $△$ ABC is $\lambda$ times the area of $△$FCE, then $\lambda$ =

• A. 1/4
• B. 4
• C. 1/2
• D. 2

Answer 1

The correct option is B

4

Let A be the origin. $\overline{AB}=\overline{b},\overline{AC}=\overline{c}$

Area of $△ABC=\frac{1}{2}(\overline{b}\times \overline{c})$

$\overline{AF}=\frac{\overline{b}}{2},\overline{AE}=\frac{\overline{c}}{2}\Rightarrow \overline{FE}=\frac{\overline{c}}{2}-\frac{\overline{b}}{2},\overline{FC}=\overline{c}-\frac{\overline{b}}{2}$

area of $△FCE=\frac{1}{2}(\frac{\overline{c}}{2}-\frac{\overline{b}}{2})\times (\overline{c}-\frac{\overline{b}}{2})=\frac{1}{8}|\overline{b}\times \overline{c}|=\frac{1}{4}.△ABC$