Question

1. find the value of the variable.

12 ft
9 ft
3.5
7
3
1.5

Explanation:

Step1: Recall the tangent - secant theorem

The tangent - secant theorem states that if a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external part. Let the radius of the circle be \(a\), the length of the tangent is \(12\) ft, the external part of the secant is \(9\) ft, and the entire secant segment is \(9 + 2a\) (since the diameter is \(2a\)). So by the tangent - secant theorem, we have \(12^{2}=9\times(9 + 2a)\).

Step2: Simplify the equation

First, calculate \(12^{2}=144\). So the equation becomes \(144=81 + 18a\). Then, subtract \(81\) from both sides: \(144-81=18a\), which simplifies to \(63 = 18a\).

Step3: Solve for \(a\)

To find \(a\), divide both sides of the equation \(63 = 18a\) by \(18\). So \(a=\frac{63}{18}=\frac{7}{2}=3.5\)? Wait, no, wait. Wait, maybe I made a mistake. Wait, the secant segment: the external part is \(9\), and the internal part is the diameter? Wait, no, the secant goes through the center, so the length from the external point \(C\) to the far end of the secant is \(9 + 2a\), and the tangent is \(12\). But let's re - check. Wait, the formula is \(tangent^{2}=external\times(internal + external)\). So \(12^{2}=9\times(9 + 2a)\). \(144 = 81+18a\). \(18a=144 - 81=63\). \(a=\frac{63}{18}=3.5\)? But wait, the options have \(3.5\), \(7\), \(3\), \(1.5\). Wait, maybe I misread the diagram. Wait, maybe the secant is \(9 + a\) and the other part is \(a\)? Wait, no, the diameter is \(2a\), so the secant is composed of the external part \(9\) and the internal part \(2a\). Wait, maybe the diagram is such that the distance from \(C\) to the center is \(9 + a\), and the radius is \(a\), and the tangent is \(12\). Then by the Pythagorean theorem, because the tangent is perpendicular to the radius, so \(12^{2}+a^{2}=(9 + a)^{2}\). Let's try that. Expand \((9 + a)^{2}=81 + 18a+a^{2}\). Then \(144+a^{2}=81 + 18a+a^{2}\). Subtract \(a^{2}\) from both sides: \(144=81 + 18a\). \(18a=63\), \(a = 3.5\). Yes, that's the same as before. So \(a = 3.5\).

Answer:

\(3.5\) (corresponding to the option with value \(3.5\))