Question

Assertion : Area of an equilateral triangle of side ' ${\mathrm{a}}^{\text{'}}$ is $=\frac{\sqrt{3}}{4}{\mathrm{a}}^2$.
Reason : The height of an equilateral triangle of side '$a$ ' is $=\frac{\sqrt{3}}{4}a$

Which of the following is correct?

• A. $\mathrm{A}$ is true and $\mathrm{R}$ is false.
• B. $A$ is false and $R$ is true.
• C. Both $\mathrm{A}$ and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
• D. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A$.

Answer 1

The correct option is A

$\mathrm{A}$ is true and $\mathrm{R}$ is false.

Explanation for correct option:

A:

Step 1: Assertion : Area of an equilateral triangle of side ' ${\mathrm{a}}^{\text{'}}$ is $=\frac{\sqrt{3}}{4}{\mathrm{a}}^2$.

If the side of an equilateral triangle is ' $a$ '

Then by Heron's formula.

$s=\frac{a+b+c}{2}=\frac{a+a+a}{2}=\frac{3a}{2}$

Area of the triangle
$$\begin{gathered} =\sqrt{\left.\frac{3}{2}a\left(\frac{3}{2}a-a\right)\left(\frac{3}{2}a-a\right)\right)\left(\frac{3}{2}a-a\right)} \\ =\sqrt{\frac{3}{2}a\times \frac{a}{2}\times \frac{a}{2}\times \frac{a}{2}}=\frac{\sqrt{3}}{4}{a}^2 \end{gathered}$$

Hence, assertion is correct.

Step 2: Reason : The height of an equilateral triangle of side '$a$ ' is $=\frac{\sqrt{3}}{4}a$

If height of an equilateral triangle be h,

We know that, we can find the area by using formula: $Area=\frac{1}{2}\times base\times height$

So, equating both we will get,
$\frac{1}{2}\times \mathrm{a}\times \mathrm{h}=\frac{\sqrt{3}}{4}{\mathrm{a}}^2$
Therefore,$h=\frac{\sqrt{3}}{2}a$

Hence, reason is not correct.

Hence, Assertion is true but reason is false.

Hence option A is correct.