Question

If the orthocentre, centroid, incentre and circumcentre in a $△ABC$ coincide with each other and if the length of side $AB$ is $\sqrt{75}$ units, then the length of the altitude of the triangle through the vertex $A$ is

• A. $\sqrt{3}$
• B. $3$ units
• C. $\frac{\sqrt{15}}{2}$ units
• D. $\frac{15}{2}$
• E. $\frac{\sqrt{5}}{2}$

Answer 1

Answer: (D)

$\frac{15}{2}$

Given, Length of side $AB\left(a\right)=\sqrt{75}$
since, orthocentre, centroid, incentre and circumcentre are coincide in $△ABC$
$\therefore$ $△ABC$ is an equilateral triangle
Now, length of the altitude
$=\frac{\sqrt{3}}{2}a=\frac{\sqrt{3}}{2}\times \sqrt{75}=\frac{15}{2}$ units