Question

find the value of x
5x-8
2x+16

Explanation:

Step1: Recall tangent-secant theorem

In a circle, if a tangent and a secant are drawn from an external point, the square of the tangent length equals the product of the secant segment and its external part. Here, the tangent is \(5x - 8\) and the secant's external part is \(2x + 16\) (assuming the secant's entire length is such that the external part is \(2x + 16\), but actually, wait—wait, maybe it's a tangent and a secant where the tangent length squared equals the secant's external segment times the entire secant? Wait, no, looking at the diagram, maybe it's a tangent and a secant where the tangent is \(5x - 8\) and the secant has a segment \(2x + 16\) as the external part? Wait, no, maybe the two segments are equal? Wait, no, the problem: the diagram shows a circle, a tangent from an external point with length \(5x - 8\), and a secant from the same external point passing through the circle, with the external segment (from external point to circle) being \(2x + 16\)? Wait, no, maybe it's a tangent and a secant where the tangent length is equal to the length of the secant's external part? Wait, no, the correct theorem is: if a tangent segment and a secant segment are drawn from an external point to a circle, then \( \text{tangent}^2=\text{external part of secant} \times \text{entire secant} \). But in the diagram, maybe the secant is actually a tangent? No, the diagram shows a tangent and a secant? Wait, no, maybe it's two tangents? Wait, the diagram: there's a circle, a point outside, a tangent touching the circle, and a segment from the external point to the circle (maybe a tangent? No, the labels are \(5x - 8\) (tangent) and \(2x + 16\) (another tangent? Wait, if two tangents are drawn from the same external point, they are equal in length. Ah! That's the key. The two tangent segments from a common external point to a circle are congruent. So \(5x - 8 = 2x + 16\).

Step2: Solve the equation \(5x - 8 = 2x + 16\)

Subtract \(2x\) from both sides: \(5x - 2x - 8 = 16\) → \(3x - 8 = 16\)
Add 8 to both sides: \(3x = 16 + 8\) → \(3x = 24\)
Divide by 3: \(x = \frac{24}{3} = 8\)

Wait, but let's check. If \(x = 8\), then \(5x - 8 = 5(8) - 8 = 40 - 8 = 32\), and \(2x + 16 = 16 + 16 = 32\). So they are equal, which makes sense for two tangents from the same external point. So that's correct.

Answer:

\(x = 8\)