Question

in circle m, or = 6 and vt = 5.
find x, the length of \overline{tr}.

Explanation:

Step1: Recall the tangent-secant theorem

The tangent-secant theorem states that if a tangent segment \( OR \) and a secant segment \( RV \) are drawn from an external point \( R \) to a circle, then \( OR^2 = RT \times RV \). Let \( RT = x \) and \( RV = RT + TV = x + 5 \).

Step2: Apply the theorem

Substitute the known values into the formula: \( 6^2 = x(x + 5) \). So, \( 36 = x^2 + 5x \). Rearrange this to a quadratic equation: \( x^2 + 5x - 36 = 0 \).

Step3: Solve the quadratic equation

Factor the quadratic equation: \( x^2 + 5x - 36=(x + 9)(x - 4)=0 \). Setting each factor equal to zero gives \( x + 9 = 0 \) or \( x - 4 = 0 \). Since length cannot be negative, we discard \( x=-9 \). So, \( x = 4 \).

Answer:

\( 4 \)