Multiple choice

$ABCD$ is a square inscribed in a circle of radius $14$ cm. $E, F, G$ and $H$ are the midpoints of the sides $DA, AB, BC$ and $CD$ respectively. The area of the square $EFGH$ will be equal to

  1. $89$ $\displaystyle cm^{2}$
  2. $196$ $\displaystyle cm^{2}$
  3. $98$ $\displaystyle cm^{2}$
  4. $392$ $\displaystyle cm^{2}$
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B Correct answer
Explanation

Square ABCD inscribed in circle radius 14. Diagonal of square = diameter = 28. Side of ABCD = 28 / sqrt(2) = 14 * sqrt(2). Area of ABCD = 196 * 2 = 392. EFGH is formed by midpoints, so its area is half of ABCD. Area = 392 / 2 = 196.

AI explanation

The diagonal of the outer square ABCD is the diameter of the circle, which is 2 times 14 equal to 28 cm. Using the Pythagorean theorem, the side length of ABCD is 28 divided by the square root of 2, which is 14 times the square root of 2. E, F, G, and H are midpoints, so the area of inner square EFGH is half the area of ABCD. The area of ABCD is 14 squared times 2 which is 392, and half of that is 196 cm^2.

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