Question

in the accompanying diagram, \\( \overline{pt} \\) is tangent to circle \\( o \\) at \\( t \\) and \\( \overline{pkm} \\) is a secant. if \\( pk = 4 \\) and \\( pm = 9 \\), find \\( pt \\).

Explanation:

Step1: Apply the Tangent-Secant Rule

The tangent-secant rule states that if a tangent segment \( PT \) and a secant segment \( PKM \) are drawn from an external point \( P \) to a circle, then \( PT^2 = PK \times PM \).
Given \( PK = 4 \) and \( PM = 9 \), substitute these values into the formula:
\( PT^2 = 4 \times 9 \)

Step2: Calculate \( PT^2 \)

First, calculate the product \( 4 \times 9 \):
\( PT^2 = 36 \)

Step3: Solve for \( PT \)

Take the square root of both sides to find \( PT \). Since length cannot be negative, we take the positive square root:
\( PT = \sqrt{36} = 6 \)

Answer:

\( 6 \)