Question

there is a geometric figure with a circle and a triangle. the triangle has a side labeled ( x + 4 ) (a tangent to the circle) and another side labeled ( x ) (a secant to the circle, with the part inside the circle being 10). the multiple - choice options are 8, 11, 6, 9.

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. Here, the tangent length is \(x + 4\), the external part of the secant is \(x\), and the entire secant length is \(x + 10\) (since the internal part of the secant is 10). So we have the equation \((x + 4)^2=x(x + 10)\).

Step2: Expand and simplify the equation

Expand the left - hand side: \((x + 4)^2=x^{2}+8x + 16\).
The right - hand side is \(x(x + 10)=x^{2}+10x\).
So the equation becomes \(x^{2}+8x + 16=x^{2}+10x\).

Step3: Solve for x

Subtract \(x^{2}\) from both sides of the equation: \(8x + 16 = 10x\).
Subtract \(8x\) from both sides: \(16=10x - 8x\), which simplifies to \(2x = 16\).
Divide both sides by 2: \(x=\frac{16}{2}=8\).

Answer:

8