The image shows a circle with center $O$, and two radii $OA$ and $OB$ forming a triangle $\triangle ABO$. The length of $OA = OB = 10 \, \text{cm}$, and the distance between points $A$ and $B$ is given as 12 cm.

To calculate the area of $\triangle ABO$:

1. Identify the necessary elements:

   • $OA = OB = 10 \, \text{cm}$ (the radii of the circle),
   • $AB = 12 \, \text{cm}$ (chord between $A$ and $B$).

2. Use Heron's formula for the area of a triangle: Heron's formula is used when we know the lengths of all three sides of the triangle. It is given by: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s$ is the semi-perimeter, and $a$, $b$, and $c$ are the sides of the triangle.

   Here:

   • $a = 10 \, \text{cm}$ (side $OA$),
   • $b = 10 \, \text{cm}$ (side $OB$),
   • $c = 12 \, \text{cm}$ (side $AB$).

   The semi-perimeter $s$ is: $s = \frac{a + b + c}{2} = \frac{10 + 10 + 12}{2} = 16 \, \text{cm}$

3. Substitute the values into Heron’s formula: $A = \sqrt{16(16 - 10)(16 - 10)(16 - 12)}$ $A = \sqrt{16 \times 6 \times 6 \times 4} = \sqrt{16 \times 144} = \sqrt{2304} = 48 \, \text{cm}^2$

Thus, the area of $\triangle ABO$ is 48 cm².

Would you like more details or have any further questions?

Related questions:

1. How do we derive Heron’s formula for a triangle’s area?
2. What other methods can be used to find the area of a triangle besides Heron's formula?
3. How does the area change if the length of the chord $AB$ is reduced?
4. Can we use trigonometry to find the area of $\triangle ABO$?
5. What is the formula for the area of a sector in a circle, and how is it related to this problem?

Tip: When solving triangle problems, if all three sides are known, Heron's formula is often the most straightforward method to calculate the area.