Question

If points $D, E$ and $F$ divide sides $BC, CA$ and $AB$ respectively in ratio $\lambda:1$ (in order) and or $ar(\triangle DEF)=0.4$ $ar(\triangle ABC)$, then $\lambda$ is equal to :

• A. $\frac{2-\sqrt{5}}{2}$
• B. $\frac{3-\sqrt{5}}{2}$
• C. $\frac{2+\sqrt{5}}{2}$
• D. $\frac{3+\sqrt{5}}{2}$

Answer 1

Answer: (B), (D)

$\frac{3-\sqrt{5}}{2}$ and $\frac{3+\sqrt{5}}{2}$

Answer 2

The problem asks us to find the value of $\lambda$ given that points $D, E, F$ divide sides $BC, CA, AB$ respectively in the ratio $\lambda:1$, and the area of $\triangle DEF$ is $0.4$ times the area of $\triangle ABC$.

Let $ar(\triangle ABC)$ denote the area of triangle $ABC$. The ratio of the area of the inner triangle $DEF$ to the area of the outer triangle $ABC$, when the sides are divided in the ratio $m:n$ (i.e., $BD:DC = m:n$, $CE:EA = m:n$, $AF:FB = m:n$), is given by the formula:

$$\frac{ar(\triangle DEF)}{ar(\triangle ABC)} = \frac{m^2 - mn + n^2}{(m+n)^2}$$

In this problem, the ratio is $\lambda:1$, so we have $m=\lambda$ and $n=1$. Substituting these values into the formula, we get:

$$\frac{ar(\triangle DEF)}{ar(\triangle ABC)} = \frac{\lambda^2 - \lambda \cdot 1 + 1^2}{(\lambda+1)^2} = \frac{\lambda^2 - \lambda + 1}{(\lambda+1)^2}$$

We are given that $ar(\triangle DEF) = 0.4 \cdot ar(\triangle ABC)$. This means $\frac{ar(\triangle DEF)}{ar(\triangle ABC)} = 0.4 = \frac{4}{10} = \frac{2}{5}$.

Now, we equate the two expressions for the ratio of the areas:

$$\frac{\lambda^2 - \lambda + 1}{(\lambda+1)^2} = \frac{2}{5}$$

Cross-multiply to solve for $\lambda$:

$$5(\lambda^2 - \lambda + 1) = 2(\lambda+1)^2$$ $$5\lambda^2 - 5\lambda + 5 = 2(\lambda^2 + 2\lambda + 1)$$ $$5\lambda^2 - 5\lambda + 5 = 2\lambda^2 + 4\lambda + 2$$

Rearrange the terms to form a quadratic equation:

$$(5\lambda^2 - 2\lambda^2) + (-5\lambda - 4\lambda) + (5 - 2) = 0$$ $$3\lambda^2 - 9\lambda + 3 = 0$$

Divide the entire equation by 3:

$$\lambda^2 - 3\lambda + 1 = 0$$

Now, use the quadratic formula $\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the values of $\lambda$. Here, $a=1$, $b=-3$, $c=1$.

$$\lambda = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(1)}}{2(1)}$$ $$\lambda = \frac{3 \pm \sqrt{9 - 4}}{2}$$ $$\lambda = \frac{3 \pm \sqrt{5}}{2}$$

This gives two possible values for $\lambda$:

1. $\lambda_1 = \frac{3 + \sqrt{5}}{2}$
2. $\lambda_2 = \frac{3 - \sqrt{5}}{2}$

Both values are positive ($\sqrt{5} \approx 2.236$), so both represent valid ratios for internal division of the sides.

Both $\frac{3-\sqrt{5}}{2}$ and $\frac{3+\sqrt{5}}{2}$ are present in the given options (B) and (D) respectively.

Therefore, both (B) and (D) are correct.