b. 12

UpStudy Step by Step Solution:

To solve for \(x\)given that \(AB\)and \(AC\) are tangent to the circle from the same external point \(A\), we can use the property of tangents from a common external point. The lengths of the tangents drawn from an external point to a circle are equal.

Given:

- \(AB = 45\)

- \(AC = 3x + 9\)

Since \(AB\) and \(AC\) are tangents from point \(A\):

\(AB = AC\)

Therefore:

\(45 = 3x + 9\)

To solve for \(x\):

1. Subtract 9 from both sides:

\(45 - 9 = 3x\)

\(36 = 3x\)

2. Divide both sides by 3:

\(x = \frac { 36} { 3} \)

\(x = 12\)

So, the value of \(x\) is \(12\).

Supplemental Knowledge:

In geometry, a fundamental property of tangents to a circle is that the lengths of two tangents drawn from an external point to the circle are equal. This is known as the tangent-segment theorem. If \(A\) is an external point and \(AB\) and \(AC\) are tangents to a circle from point \(A\), then \(AB = AC\).

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