Multiple choice

In a right-angled triangle ABC, angle A is 90 degrees. The lengths of sides AC and AB are 6 cm and 8 cm, respectively. A circle is inscribed within the triangle, with sides AB, BC, and AC as tangents to the inscribed circle. Another circle is circumscribed around the triangle, passing through points A, B, and C. Find the ratio of the area of the inscribed circle to the area of the circumscribed circle.

  1. 1 : 3

  2. 2 : 5

  3. 9 : 16

  4. 4 : 25

  5. a
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D Correct answer
Explanation

In a 6-8-10 triangle, the inradius r = (a+b-c)/2 = (6+8-10)/2 = 2. The circumradius R = hypotenuse/2 = 10/2 = 5. The ratio of areas is (pi*r^2) / (pi*R^2) = r^2 / R^2 = 2^2 / 5^2 = 4/25.

AI explanation

Since angle A is 90 degrees, BC is the hypotenuse, and by the Pythagorean theorem its length is the square root of (6 squared plus 8 squared), which is 10 centimeters. The radius of the circumscribed circle is half the hypotenuse, so it is 5 centimeters. The radius of the inscribed circle is found by the formula the area divided by the semi-perimeter, giving an inscribed radius of (24 divided by 12), which is 2 centimeters. The ratio of the areas of the two circles is the square of the ratio of their radii, resulting in 2 squared to 5 squared, or a ratio of 4 to 25.

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