Question : If the area of an equilateral triangle is $a$ and height $b$, then the value of $\frac{b^2}{a}$ is:
Option 1: $3$
Option 2: $\frac{1}{3}$
Option 3: $\sqrt3$
Option 4: $\frac{1}{\sqrt3}$
Correct Answer: $\sqrt3$
Solution : The Area of the equilateral triangle = $\frac12×\ \text{base × height}$ $⇒a= \frac12×\ \text{base}\ ×b$ $\therefore$ base $= \frac{2a}{b}$ Now, Area of an equilateral triangle $= \frac{\sqrt3}{4}\times (\text{side})^2$ $⇒a= \frac{\sqrt3}{4}\times (\frac{2a}{b})^2$ $⇒\frac{b^2}{a} = \sqrt3$ Hence, the correct answer is $\sqrt3$.
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Question : ABCD is a square. Draw an equilateral $\triangle $PBC on side BC considering BC is a base and an equilateral $\triangle $QAC on diagonal AC considering AC is a base. Find the value of $\frac{\text{area of $\triangle PBC$}}{\text{area of $\triangle QAC$}}$.
Question : The altitude of an equilateral triangle of side $\frac{2}{\sqrt3}$ cm is:
Question : If the numerical values of the height and the area of an equilateral triangle are the same, then the length of each side of the triangle is:
Question : If $\frac{\tan\theta +\cot\theta }{\tan\theta -\cot\theta }=2, (0\leq \theta \leq 90^{0})$, then the value of $\sin\theta$ is:
Question : If $N=\frac{(\sqrt7-\sqrt3)}{(\sqrt7+\sqrt3)}$, then what is the value of $(N+\frac{1}{N})$?
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