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In an equilateral triangle $$ABC$$, $$AD$$ is an altitude. Then $$4AD^{2}$$ is equal to（ ）
A. $$2BD^2$$
B. $$BC^2$$
C. $$3AB^2$$
D. $$2DC^2$$
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## QuestionMathsClass 7

In an equilateral triangle $$ABC$$, $$AD$$ is an altitude. Then $$4AD^{2}$$ is equal to（ ）
A. $$2BD^2$$
B. $$BC^2$$
C. $$3AB^2$$
D. $$2DC^2$$

C
4.6
4.6

## Solution

In an equilateral triangle, all sides are equal.

In $$\Delta ADC$$, applying Pythagoras theorem, we get
$$AD^2=\sqrt{AC^2-DC^2}$$
As, $$DC=\frac{BC}{2}$$
$$\Rightarrow$$  $$AD=\sqrt{AC^2-(\frac{BC}{2})^2}$$
$$\Rightarrow$$  $$AD=\sqrt{AC^2-\frac{BC^2}{4}}$$
$$\Rightarrow$$  $$AD=\sqrt{\frac{4\ AC^2-BC^2}{4}}$$
As all sides are equal i.e. $$AB=BC=CA$$
$$\Rightarrow$$  $$AD=\sqrt{\frac{4\ AB^2-AB^2}{4}}$$
$$\Rightarrow$$  $$AD^2=\frac{4\ AB^2-AB^2}{4}$$
$$\Rightarrow$$  $$4AD^2=3\ AB^2$$