Multiple choice

What is the area of the triangle in which two of its medians $9$ cm and $12$ cm long intersect at right angles?

  1. $64$ sq. cm
  2. $68$ sq. cm
  3. $72$ sq. cm
  4. $86$ sq. cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a triangle with medians m1 and m2 intersecting at right angles, the area is (2/3) * m1 * m2. Here, (2/3) * 9 * 12 = 72.

AI explanation

The centroid divides the medians in a 2:1 ratio, so the segments of the 9 cm median are 6 cm and 3 cm, and the segments of the 12 cm median are 8 cm and 4 cm. Because the medians intersect at right angles, the total area of the triangle is the sum of the areas of the three small right triangles plus the central one, or six times the area of the smallest triangle formed by the shorter segments. Calculating 0.5 times (6 times 4 + 6 times 8 + 3 times 8) gives 0.5 times 24 plus 0.5 times 48 plus 0.5 times 24, resulting in a total area of 72 sq. cm.

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