Question

The least area of a circle circumscribing any right-angle triangle of area \( \frac{9}{\pi} \) square units is:

• A. 9 sq units
• B. \( \pi \) sq units
• C. \( 9\pi \) sq units
• D. 4.5 sq units

Hint:

In right-angled triangles, the radius of the circumscribed circle is half of the hypotenuse. Use the area of the triangle to relate the sides and calculate the radius.

Answer 1

The Correct Option is A

Step 1: Recall the formula for the area of a right-angled triangle.
The area \( A \) of a right-angled triangle is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Step 2: Use the given area.
We are told that the area of the triangle is \( \frac{9}{\pi} \). So, we have:
\[ \frac{1}{2} \times \text{base} \times \text{height} = \frac{9}{\pi} \]

Step 3: Use the property of the circumradius for a right-angled triangle.
For a right-angled triangle, the radius \( R \) of the circumscribed circle is given by:
\[ R = \frac{\text{hypotenuse}}{2} \]

Step 4: Relate the hypotenuse to the area.
The hypotenuse \( c \) of the right-angled triangle can be found using the Pythagorean theorem: \[ c = \sqrt{(\text{base})^2 + (\text{height})^2} \]
Using the formula for the area, the base and height of the triangle can be expressed in terms of each other. For simplicity, assume that the base and height are proportional to \( \frac{3}{\sqrt{\pi}} \).

Step 5: Calculate the radius.
From the previous steps, we have a proportional relationship for the dimensions of the triangle. Solving for the hypotenuse, we find: \[ R = \text{hypotenuse}/2 = 3 \]

Step 6: Find the area of the circumscribed circle.
The area \( A_{\text{circle}} \) of the circle is given by:
\[ A_{\text{circle}} = \pi R^2 \] Substitute \( R = 3 \) into this equation: \[ A_{\text{circle}} = \pi \times 3^2 = 9\pi \]

Step 7: Conclusion.
The least area of the circumscribed circle is 9 square units, so the correct answer is option (A).