To solve this problem, we need to use the Power of a Point theorem. Given that [tex]\( AT \)[/tex] is a secant and [tex]\( PT \)[/tex] is a tangent:
1. Step 1: Identify the given values:
- Length of the secant [tex]\( AT = 9 \)[/tex] cm
- Length of the chord [tex]\( BP = 4 \)[/tex] cm
2. Step 2: Understand the relationship between the lengths using the Power of a Point theorem. According to this theorem for secants and tangents:
[tex]\[
PT^2 = AT \times AP
\][/tex]
where [tex]\( AP \)[/tex] is the remaining part of the secant.
3. Step 3: Calculate the length [tex]\( AP \)[/tex]:
[tex]\[
AP = AT - BP
\][/tex]
Substituting the given values:
[tex]\[
AP = 9 \, \text{cm} - 4 \, \text{cm} = 5 \, \text{cm}
\][/tex]
4. Step 4: Applying the Power of a Point theorem:
[tex]\[
PT^2 = AT \times AP
\][/tex]
Substituting [tex]\( AT \)[/tex] and [tex]\( AP \)[/tex]:
[tex]\[
PT^2 = 9 \, \text{cm} \times 5 \, \text{cm} = 45 \, \text{cm}^2
\][/tex]
5. Step 5: Find the length [tex]\( PT \)[/tex] by taking the square root of 45:
[tex]\[
PT = \sqrt{45} \approx 6.708 \, \text{cm}
\][/tex]
After doing these calculations, we find:
[tex]\[
AP = 5 \, \text{cm}, \, PT^2 = 45 \, \text{cm}^2, \, PT \approx 6.708 \, \text{cm}
\][/tex]
Since the length of the chord [tex]\( BP = 4 \, \text{cm} \)[/tex] is given directly in the problem, you can check your result to confirm BP is accurate.
Therefore, the correct length of the chord [tex]\( BP \)[/tex] is:
[tex]\[ a. 5 \, \text{cm} \][/tex]
