Question

question 2
determine the value of x.

Explanation:

Step1: Recall the secant-secant theorem

The secant-secant theorem states that if two secant segments are drawn from a point outside a circle, then the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part. The formula is \( (external\ part + internal\ part) \times external\ part=(external\ part' + internal\ part') \times external\ part' \). For the given diagram, let the external part of the first secant be \( x \) and internal part be \( 1 \), and for the second secant, external part is \( 2 \) and internal part is \( 4 \). So the formula becomes \( x(x + 1)=2(2 + 4) \).

Step2: Simplify the equation

First, calculate the right - hand side: \( 2(2 + 4)=2\times6 = 12 \). So our equation is \( x^{2}+x - 12=0 \).

Step3: Solve the quadratic equation

We factor the quadratic equation \( x^{2}+x - 12 = 0 \). We need two numbers that multiply to \( - 12 \) and add up to \( 1 \). The numbers are \( 4 \) and \( - 3 \). So \( (x + 4)(x - 3)=0 \). Setting each factor equal to zero gives \( x=-4 \) or \( x = 3 \). Since length cannot be negative, we discard \( x=-4 \).

Answer:

\( x = 3 \)