Understanding the Circumcircle of a Right-Angled Triangle

The problem asks us to find the area of the circumcircle of a right-angled triangle given the lengths of its two sides that form the right angle. These sides are also known as the legs or cathetus of the right triangle.

In a right-angled triangle, a special property of the circumcircle is that its diameter is equal to the length of the hypotenuse of the triangle. The circumcenter (the center of the circumcircle) is located exactly at the midpoint of the hypotenuse.

Steps to Solve the Problem

To find the area of the circumcircle, we first need to determine its radius. Since the diameter is the hypotenuse, the radius will be half the length of the hypotenuse.

Step 1: Calculate the Hypotenuse

We are given the lengths of the two legs holding the right angle as 3 cm and 4 cm. We can use the Pythagorean theorem to find the length of the hypotenuse (let's call it $c$). The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs). Let the legs be $a$ and $b$.

The formula is: $a^2 + b^2 = c^2$

Substitute the given values:

$ (3 \text{ cm})^2 + (4 \text{ cm})^2 = c^2 $

$ 9 \text{ cm}^2 + 16 \text{ cm}^2 = c^2 $

$ 25 \text{ cm}^2 = c^2 $

Taking the square root of both sides:

$ c = \sqrt{25 \text{ cm}^2} $

$ c = 5 \text{ cm} $

So, the length of the hypotenuse is 5 cm.

Step 2: Determine the Circumradius

As mentioned, the diameter of the circumcircle of a right triangle is the hypotenuse. The circumradius (let's call it $R$) is half of the diameter.

$ R = \text{Hypotenuse} / 2 $

$ R = 5 \text{ cm} / 2 $

$ R = 2.5 \text{ cm} $

The circumradius is 2.5 cm.

Step 3: Calculate the Area of the Circumcircle

The area of a circle is given by the formula: Area $= \pi R^2$, where $R$ is the radius.

Substitute the calculated circumradius:

Area $= \pi (2.5 \text{ cm})^2 $

Area $= \pi (2.5 \times 2.5) \text{ cm}^2 $

Area $= \pi (6.25) \text{ cm}^2 $

Area $= 6.25\pi \text{ cm}^2 $

Thus, the area of the circumcircle is $6.25\pi \text{ cm}^2$. Looking at the options provided, this matches one of them.

Conclusion

By calculating the hypotenuse of the right-angled triangle and using it to find the circumradius, we determined the area of the circumcircle. The calculated area is $6.25\pi \text{ cm}^2$.

Revision Table: Key Concepts for Circumcircles

Additional Information on Triangle Circumcircles

The circumcircle exists for every triangle. The location of the circumcenter depends on the type of triangle:

• For an acute triangle (all angles < 90°), the circumcenter lies inside the triangle.
• For a right-angled triangle, the circumcenter lies on the midpoint of the hypotenuse.
• For an obtuse triangle (one angle > 90°), the circumcenter lies outside the triangle.

The circumradius formula $R = \frac{abc}{4K}$ is a general formula for any triangle, where $a, b, c$ are the lengths of the sides, and $K$ is the area of the triangle. For a right triangle with legs $a$ and $b$, the area $K = \frac{1}{2}ab$. The hypotenuse $c$ can be found using $c = \sqrt{a^2 + b^2}$. Substituting these into the general formula:

$ R = \frac{ab\sqrt{a^2+b^2}}{4(\frac{1}{2}ab)} = \frac{ab\sqrt{a^2+b^2}}{2ab} = \frac{\sqrt{a^2+b^2}}{2} $

Since $\sqrt{a^2+b^2}$ is the hypotenuse, this confirms that for a right triangle, $R = \frac{\text{hypotenuse}}{2}$. This specific property simplifies finding the circumradius and subsequently the area of the circumcircle for right-angled triangles.