Circle Geometry, SMB-002

The line \(AT\) is the tangent to the circle at \(A\), and \(BT\) is a secant meeting the circle at \(B\) and \(C\).

Given that \(AT = 12\), \(BC = 7\) and \(CT = x\), find the value of \(x\). (2 marks)

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\(x = 9\)

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\(\text{Property: square of tangent = product of secant intercepts}\)

\(x^2 + 7x-144\)	\(= 0\)
\((x + 16)(x-9)\)	\(= 0\)

\(\therefore x = 9,\ (x \gt 0) \)