Multiple choice

$OABC$ is a rhombus whose three vertices $A, B$ and $C$ lie on a circle with centre $O$. If the radius of the circle is $10$ cm. Find the area of the rhombus.

  1. $25$ $\sqrt{3}$ sq. cm.
  2. $50$ $\sqrt{3}$ sq. cm.
  3. $75$ $\sqrt{3}$ sq. cm.
  4. $20$ $\sqrt{3}$ sq. cm.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a rhombus OABC with O as the center, the sides OA, OB, OC are radii (10 cm). The triangles OAB and OCB are equilateral triangles. The area is 2 * (sqrt(3)/4 * 10^2) = 50 * sqrt(3).

AI explanation

Since A, B, and C lie on the circle with center O and OA and OC are radii of 10 centimeters, the diagonals of the rhombus OABC are OB and AC. Angle AOC is 120 degrees and the diagonals are perpendicular, meaning the area of the rhombus is half the product of its diagonals. Diagonal OB is 10 centimeters, and diagonal AC is composed of two right triangle legs opposite 60 degree angles, giving 2 times 10 times the square root of 3, which is 20 times the square root of 3. Therefore, the area is half of 10 times 20 times the square root of 3, resulting in 50 times the square root of 3 square centimeters.

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