# Ex.6.4 Q8 Triangles Solution - NCERT Maths Class 10

## Question

\(ABC\) and \(BDE\) are two equilateral triangles such that \(D\) is the mid-point of \(BC.\) Ratio of the areas of triangles \(ABC\) and \(BDE\) is

(a) \(2 : 1\)

(b) \(1 : 2\)

(c) \(4 : 1\)

(d) \(1 : 4\)

## Text Solution

**Reasoning:**

**\(AAA\) **criterion.

**Steps: **

\(\Delta A B C \sim \Delta B D E\) \((\because \text{equilateral triangles)}\)

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

\[\begin{aligned} \frac{ar\Delta A B C}{ar \Delta B D E} &=\frac{(B C)^{2}}{(B D)^{2}} \\ &=\frac{(B C)^{2}}{\left(\frac{B C}{2}\right)^{2}} \\ [ D \text { is the} & \text{ mid point} \text{ of } BC] \\ &=\frac{(B C)^{2} \times 4}{(B C)^{2}}\\&=4 \end{aligned}\]

\[\begin{aligned} ar\Delta ABE : ar\Delta BDE =4:1\end{aligned}\]

The answer is (c) \( 4:1\)