Multiple choice

$s_{1}$ and $s_{2}$ are the inscribed and circumscribed circles of a triangle with sides, $3, 4$ and $5$, Then (area of $s_{1}$/ area of $s_{2}$)=

  1. $16/25$
  2. $4/25$
  3. $9/25$
  4. $9/16$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a right triangle with sides 3, 4, 5, the inradius r = (a+b-c)/2 = (3+4-5)/2 = 1. The circumradius R = c/2 = 5/2 = 2.5. Area of inscribed circle = pi * r^2 = pi. Area of circumscribed circle = pi * R^2 = 6.25 * pi. Ratio = 1 / 6.25 = 1 / (25/4) = 4/25.

AI explanation

The semi-perimeter of the triangle is 6, and using Heron's formula, the area is the square root of (6 times 3 times 2 times 1), which equals 6. The inradius is the area divided by the semi-perimeter, giving 1, so the area of the inscribed circle is pi. The hypotenuse of 5 serves as the circumradius, making the circumscribed circle area 25 times pi. The ratio of these two areas is pi divided by 25 times pi, which equals 4/25.

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