`XY and XZ `are tangents to a circle, `ST `is another tangent to the circle at the point R on the circle, which intersects `XY and XZ `at `S and T` respectively. If `XY = 15 cm and TX = 9 cm`, then `RT `is

Options

A. 4.5 cm
B. 7.5 cm
C. 6 cm
D. 3 cm

Answer

To solve the problem step by step, we will use the properties of tangents to a circle.

### Step-by-Step Solution:

1. **Identify the Tangents**: We have two tangents, XY and XZ, which are both drawn from point X to the circle. We are given that XY = 15 cm.

2. **Use the Tangent Property**: According to the property of tangents from an external point to a circle, the lengths of the tangents drawn from a point outside the circle to the points of tangency are equal. Therefore, we have:
\[ XY = XZ = 15 \text{ cm} \]

3. **Determine TX**: We are given that TX = 9 cm.

4. **Express XZ in Terms of TX and TZ**: Since XZ can be expressed as:
\[ XZ = TX + TZ \]
Substituting the known values:
\[ 15 = 9 + TZ \]

5. **Solve for TZ**: Rearranging the equation gives us:
\[ TZ = 15 - 9 = 6 \text{ cm} \]

6. **Use the Tangent Property at Point T**: Now, since TR is also a tangent from point T to the circle, we can apply the same tangent property. Therefore:
\[ TZ = TR \]
Thus, we have:
\[ TR = 6 \text{ cm} \]

7. **Final Result**: Therefore, the length of RT is:
\[ RT = 6 \text{ cm} \]

### Conclusion:
The length of RT is **6 cm**.