Question

1. find the value of the variable.

3
7
1.5
3.5

Explanation:

Step1: Apply 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 \((9 + 2a)\) ft. So we have the equation \(12^{2}=9\times(9 + 2a)\).

Step2: Solve the equation

First, calculate \(12^{2}=144\). The equation becomes \(144 = 81+18a\).
Subtract \(81\) from both sides: \(144 - 81=18a\), which simplifies to \(63 = 18a\).
Then, solve for \(a\) by dividing both sides by \(18\): \(a=\frac{63}{18}=\frac{7}{2} = 3.5\). Wait, no, wait, I made a mistake. Wait, the secant: the diameter is \(2a\), so the entire secant is \(9 + 2a\)? Wait, no, the external part is \(9\), and the internal part is \(2a\) (since the diameter is \(2a\)). Wait, no, the tangent - secant formula is \(t^{2}=e\times s\), where \(t\) is tangent, \(e\) is external, \(s\) is the entire secant (external + internal). So \(t = 12\), \(e=9\), \(s=9 + 2a\). 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. Wait, but let me check again. Wait, maybe I misread the diagram. Wait, the secant: the distance from the external point to the center is \(9 + a\)? No, no. Wait, the tangent is perpendicular to the radius at the point of contact. So we have a right triangle with legs \(12\) (tangent) and \(a\) (radius)? No, no. Wait, no, the length from the external point \(C\) to the center is \(9 + a\), the radius is \(a\), and the tangent is \(12\). So by Pythagoras: \((9 + a)^{2}=12^{2}+a^{2}\). Ah! That's the correct approach. I used the wrong theorem earlier. The correct theorem is the Pythagorean theorem in the right triangle formed by the tangent, the radius, and the line from the external point to the center. So the right triangle has hypotenuse \(9 + a\), one leg \(12\) (tangent), and the other leg \(a\) (radius). So \((9 + a)^{2}=12^{2}+a^{2}\).
Let's expand the left - hand side: \(81+18a+a^{2}=144 + a^{2}\).
Subtract \(a^{2}\) from both sides: \(81 + 18a=144\).
Subtract \(81\) from both sides: \(18a=144 - 81 = 63\).
Then \(a=\frac{63}{18}=3.5\). Wait, but that gives \(a = 3.5\). But let's check the options. The options are 3,7,1.5,3.5. So 3.5 is an option. Wait, but maybe I made a mistake in the right triangle. Wait, the length from \(C\) to the center is \(9 + a\), the radius is \(a\), and the tangent is \(12\). So by Pythagoras: \((9 + a)^{2}=12^{2}+a^{2}\). Expanding: \(81+18a+a^{2}=144 + a^{2}\). Cancel \(a^{2}\): \(81 + 18a=144\). \(18a = 63\). \(a = 3.5\). Yes, that's correct.

Answer:

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