Multiple choice

Consider a rhombus with diagonals of length $6$ and $8$ and the inscribed circle $C_{1}$. The vertices of the rhombus are the midpoints of the sides of a rectangle, inscribed in a circle $C_{2}$. The ratio of the radius of $C_{1}$ and the radius of $C_{2}$ is

  1. Less than $0.48$
  2. $0.48$
  3. $0.56$
  4. $0.64$
  5. Greater than $0.64$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Rhombus diagonals 6 and 8 imply side = sqrt(3^2 + 4^2) = 5. Area = 0.5*6*8 = 24. Inradius r1 = Area / semi-perimeter = 24 / 10 = 2.4. The rhombus vertices are midpoints of a rectangle, so the rectangle sides are 6 and 8. The circumradius of this rectangle is R2 = sqrt(3^2 + 4^2) = 5. Ratio r1/R2 = 2.4/5 = 0.48.

AI explanation

The area of the rhombus is half the product of its diagonals, so (6 times 8) divided by 2 is 24. Since the side of the rhombus is the square root of (3 squared + 4 squared), which is 5, the inradius is the area divided by the semi-perimeter, yielding 24 divided by 10 for an inradius of 2.4. The circle C2 circumscribes a rectangle whose sides are parallel to the rhombus diagonals, meaning its diameter is the diagonal of the rhombus (10), giving a radius of 5. The ratio of the radius of C1 to the radius of C2 is 2.4 divided by 5, which equals 0.48.

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