Question : In triangle ABC which is equilateral. O is the point of intersection of altitude AL, BM, and CN. If OA = 16 cm, what is the semi-perimeter of the triangle ABC?
Option 1: $8 \sqrt 3$ cm
Option 2: $12 \sqrt 3$ cm
Option 3: $16 \sqrt 3$ cm
Option 4: $24\sqrt 3$ cm
Correct Answer: $24\sqrt 3$ cm
Solution : Given: AL, BM, and CN are altitude of $\triangle$ ABC. AL, BM, and CN are medians of $\triangle$ ABC. Since ABC is an equilateral triangle, AO = 2OL ⇒ 16 = 2OL ⇒ OL = 8 $\therefore$ AL = AO + OL = 16 + 8 = 24 cm Let AB = BC = CA = $x$ cm In $\triangle$ ALC we have, $\angle$ ALC = 90$^\circ$ ($\because$ AL is altitude) AL = 24 cm AC = $x$ cm CL = $\frac{\text{BC}}{2}$ = $\frac{x}{2}$ $\therefore x^2$ = $(\frac{x}{2})^2+24^2$ ⇒ $x^2$ = $\frac{x^2}{4}+576$ ⇒ $\frac{3x^2}{4}$ = $576$ ⇒ $x^2$ = $\frac{576×4}{3}$ ⇒ $x^2$ = $768$ ⇒ $x$ = $16\sqrt3$ Semi perimeter = $\frac{16\sqrt3+16\sqrt3+16\sqrt3}{2}$ = $\frac{48\sqrt3}{2}$ = $24\sqrt3$ Hence, the correct answer is $24\sqrt3$ cm.
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