Final answer:
The area of a right-angled triangle with the radius of its circumcircle being 3 cm and the altitude to the hypotenuse being 2 cm is 6 cm². This is calculated using the property that the altitude in such a triangle splits the hypotenuse into two equal segments whose product is the square of the altitude.
Step-by-step explanation:
To find the area of a right-angled triangle with the given parameters, one can use the fact that the hypotenuse of such a triangle is the diameter of its circumcircle.
Since the radius (r) is known to be 3 cm, the diameter (d) would be twice that, making it 6 cm. Using this diameter and the altitude (h) drawn to the hypotenuse, which is given as 2 cm, one can calculate the area (A) of the triangle using the formula for the area of a triangle A = 1/2 × base × height. Here, the base corresponds to the hypotenuse.
Considering the altitude h divides the hypotenuse into two segments, let's call them x and y. In a right-angled triangle, the product of the segments of the hypotenuse is equal to the square of the altitude drawn to the hypotenuse. Therefore, x × y = h^2.
Furthermore, in a right-angled triangle with hypotenuse as the diameter of its circumcircle, any altitude is also a median. Therefore, x = y and we have x^2 = h^2. Putting the values, we get x^2 = 2^2 = 4, hence x = y = 2.
Now we know the hypotenuse d = 6 cm splits into two equal parts by the altitude, both x and y = 2 cm. Returning to the formula for the area, we use the values for the hypotenuse as base (6 cm) and the altitude as height (2 cm) to calculate the area of the triangle:
A = 1/2 × d × h
× 1/2 × 6 cm × 2 cm = 6 cm²
The correct answer is A. 6 cm². This question involves the application of properties of the right-angled triangle, circumcircle and median, along with the basic area formula of a triangle.