Multiple choice

If the orthocentre, centroid, incentre and circumcentre in a $\triangle ABC$ 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

  1. $\sqrt { 3 } $
  2. $3$ units
  3. $\cfrac { \sqrt { 15 } }{ 2 } $ units
  4. $\cfrac { 15 }{ 2 } $
  5. $\cfrac { \sqrt { 5 } }{ 2 } $
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D Correct answer
Explanation

If orthocentre, centroid, incentre, and circumcentre coincide, the triangle is equilateral. In an equilateral triangle, the altitude is (sqrt(3)/2) * side. Altitude = (sqrt(3)/2) * sqrt(75) = (sqrt(3)/2) * 5 * sqrt(3) = (5 * 3) / 2 = 15/2.

AI explanation

Because the orthocentre, centroid, incentre and circumcentre coincide, triangle ABC is an equilateral triangle. In an equilateral triangle, the altitude h is given by the formula h = (side * sqrt(3)) / 2. Substituting AB = sqrt(75) = 5*sqrt(3) into the formula gives h = (5*sqrt(3) * sqrt(3)) / 2 = 15/2.

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