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

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## 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{\text {Area } \Delta A B C}{\text {Area } \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 midpoint of } \mathrm{BC}] \\ &=\frac{(B C)^{2} \times 4}{(B C)^{2}}\\&=4 \end{aligned}

Area of $$\,\Delta ABE$$ $$:$$ Area of$$\,\Delta BDE$$ $$= 4:1$$

The answer is (c) $$4:1$$

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