If one equilateral triangle ABC is made in a circle of radius 10 cm, then the length of a side of triangle ABC is:

Question

Accepted Answer

To determine the length of a side of an equilateral triangle inscribed in a circle with a radius of 10 cm, we can use the properties of equilateral triangles and circles.

1. **Explanation for Option 2 (Correct Answer):**

An equilateral triangle inscribed in a circle is known as a circumcircle. The center of the circle is also the centroid of the equilateral triangle. The relationship between the side length $ s $ of an equilateral triangle and the radius $ R $ of its circumcircle is given by the formula:

$$
   s = \sqrt{3} \times R
   $$

Given that the radius $ R $ of the circle is 10 cm, we can substitute this value into the formula:

$$
   s = \sqrt{3} \times 10 = 10\sqrt{3} \text{ cm}
   $$

Therefore, the length of a side of triangle ABC is $ 10\sqrt{3} $ cm, which corresponds to Option 2.

2. **Explanation for why other options are incorrect:**

- **Option 1 (10 cm):** This option suggests that the side length of the triangle is equal to the radius of the circle. However, for an equilateral triangle inscribed in a circle, the side length is not equal to the radius. The correct relationship involves the factor of $ \sqrt{3} $, as shown in the formula above.

- **Option 3 (10\sqrt{2} cm):** This option incorrectly uses the factor $ \sqrt{2} $ instead of $ \sqrt{3} $. The factor $ \sqrt{2} $ is not applicable in the context of an equilateral triangle inscribed in a circle. The correct factor is $ \sqrt{3} $.

- **Option 4 ($\frac{10}{\sqrt{3}}$ cm):** This option suggests a side length that is less than the radius of the circle, which is not possible for an equilateral triangle inscribed in a circle. The side length should be greater than the radius, as indicated by the correct formula $ s = \sqrt{3} \times R $.

In conclusion, the correct answer is Option 2, $ 10\sqrt{3} $ cm, based on the geometric properties of an equilateral triangle inscribed in a circle.