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Herons Formula — Maths STD 9 — Question

CBSE BoardEnglish MediumSTD 9MathsHerons Formula1 Mark
MCQ
Statement-1 (A): The altitude $p$ of an equilateral triangle having each side a is given by $p=\frac{\sqrt{3}}{2} a$
Statement-2 (R): The area $\Delta$ of an equilateral triangle having each side a is given by $\Delta=\frac{\sqrt{3}}{4} a^2$
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  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
✓

Answer

Correct option: B.
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
(b)
Let ABC be an equilateral triangle such that $A B=A C=B C=a$. Draw $A L \perp B C$. In triangle ALB, we obtain
$A B^2=A L^2+B L^2 \Rightarrow a^2=p^2+\frac{a^2}{4} \Rightarrow p^2=\frac{3 a^2}{4}\Rightarrow p=\frac{\sqrt{3} a}{2}$
So, statement-1 is true.
$\Delta=\frac{1}{2} \text { Base } \times \text { Height }=\frac{1}{2}(B C \times p)=\frac{1}{2}\left(a\times \frac{\sqrt{3}}{2} a\right)=\frac{\sqrt{3}}{4} a^2$
So, statement-2 is also true.
Thus, both the statements are true. Hence, option (b) is correct.

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