Multiple choice

A triangle with side lengths in the ratio 3 : 4 : 5 is inscribed in a circle of radius 3. The area of the triangle is equal to

  1. 8.64

  2. 12

  3. 6

  4. 10.28
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A Correct answer
Explanation

A triangle with sides 3:4:5 is a right-angled triangle. The hypotenuse is the diameter of the circumcircle, so diameter = 2 * radius = 6. The sides are 3k, 4k, 5k, so 5k = 6, k = 1.2. The sides are 3.6, 4.8, and 6. Area = 1/2 * base * height = 1/2 * 3.6 * 4.8 = 8.64.

AI explanation

Since the sides are in the ratio 3:4:5, the triangle is right-angled and its hypotenuse is the diameter of the circumscribed circle. Given the radius is 3, the diameter is 6, which corresponds to 5 parts of the ratio. Thus, the scaling factor is 6 divided by 5, making the other two sides 3 times 1.2 and 4 times 1.2, which are 3.6 and 4.8. The area of the right triangle is one-half times the product of the legs, so the area is 0.5 times 3.6 times 4.8, yielding 8.64.

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