Question

The sum of length of altitudes drawn from any point inside a triangle, whose all sides are equal, to all three sides is:

• A. equal to altitude of triangle
• B. 1/3 of side of triangle
• C. 1/2 of side of triangle
• D. equal to perimeter of triangle

Answer 1

The correct option is A equal to altitude of triangle

Consider an equilateral triangle ABC with each side measuring 10 units.
Therefore, the length of the altitude of the triangle willl be $\sqrt{{10}^2-5^2}$
= $5\sqrt{3}$

Consider the length of each altitude drawn from the orthocentre of the triangle to each of the sides.
Consider triangle ODC where DC = 5 units.
$\angle DOC={60}^{0}sin{60}^0=\frac{DC}{OC}=\frac{5}{OC}⟹\frac{\sqrt{3}}{2}=\frac{5}{OC}orOC=\frac{10}{\sqrt{3}}$

Hence ${OD}^2={OC}^2-{DC}^2⟹O{D}^2=\frac{100}{3}-25=\frac{25}{3}⟹OD=\frac{5}{\sqrt{3}}units\therefore sumofallaltitudes=3\ast \frac{5}{\sqrt{3}}=5\sqrt{3}units.$