Question

If the sides of $\Delta ABC$ are 5 cm, 8 cm, 9 cm, then the length of its longest altitude is equal to

• A. $\sqrt{11}cm$
• B. $\frac{5}{12}\sqrt{11}cm$
• C. $\sqrt{11}cm$
• D. $\frac{12}{5}\sqrt{11}cm$

Answer 1

The correct option is D $\frac{12}{5}\sqrt{11}cm$


Let the sides of $\Delta ABC$ be a = 5 cm, b = 8 cm and c = 9 cm.
$\therefore$ Semiperimeter, (s) = $\frac{a+b+c}{2}$
$=\frac{5+8+9}{2}$
= 11 cm
$\therefore$ Area of $\Delta ABC=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{11\times (11-5)\times (11-8)\times (11-9)}$
$=\sqrt{11\times 6\times 3\times 2}$
$=6\sqrt{11}c{m}^2$
Also, the longest altitude CD will be along the smallest side AB, i.e., 5 cm.
$\therefore$ Area of $\Delta ABC=\frac{1}{2}\times AB\times CD$
$\Rightarrow 6\sqrt{11}=\frac{1}{2}\times 5\times CD$
$\Rightarrow CD=\frac{6\sqrt{11}\times 2}{5}=\frac{12\sqrt{11}}{5}cm$
Hence, the correct answer is option (d).