The length of tangent, drawn from a point 8 cm away from the centre of a circle of radius 6 cm is :

Question

Accepted Answer

To determine the length of the tangent drawn from a point 8 cm away from the center of a circle with a radius of 6 cm, we can use the Pythagorean theorem. The tangent to a circle from an external point is perpendicular to the radius at the point of tangency. Therefore, we have a right triangle where:

- The distance from the external point to the center of the circle is the hypotenuse, which is 8 cm.
- The radius of the circle is one leg of the triangle, which is 6 cm.
- The length of the tangent is the other leg of the triangle, which we need to find.

According to the Pythagorean theorem:

$$
\text{(Hypotenuse)}^2 = \text{(Radius)}^2 + \text{(Tangent length)}^2
$$

Substituting the known values:

$$
8^2 = 6^2 + \text{(Tangent length)}^2
$$

$$
64 = 36 + \text{(Tangent length)}^2
$$

$$
64 - 36 = \text{(Tangent length)}^2
$$

$$
28 = \text{(Tangent length)}^2
$$

Taking the square root of both sides:

$$
\text{Tangent length} = \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} \text{ cm}
$$

Thus, the length of the tangent is $2\sqrt{7} \text{ cm}$, which corresponds to Option 2.

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

1. **Option 1: $\sqrt{7} \text{ cm}$** - This option is incorrect because the calculation shows that the tangent length is $2\sqrt{7} \text{ cm}$, not $\sqrt{7} \text{ cm}$.

2. **Option 3: $3\sqrt{7} \text{ cm}$** - This option is incorrect because it overestimates the tangent length. The correct calculation yields $2\sqrt{7} \text{ cm}$.

3. **Option 4: $5\sqrt{7} \text{ cm}$** - This option is incorrect because it significantly overestimates the tangent length. The correct length is $2\sqrt{7} \text{ cm}$.

Therefore, the correct answer is indeed Option 2: $2\sqrt{7} \text{ cm}$.