Question

what is the value of x? options: x=2 (marked with x), x=3, x=4, x=6 (diagram shows a circle with secant segments: ab=21, ed=14, cb=x, cd=x+1)

Explanation:

Step1: Apply 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. For secant \(CA\) with external part \(CB = x\) and the whole secant \(CA=x + 21\), and secant \(CE\) with external part \(CD=x + 1\) and the whole secant \(CE=(x + 1)+14=x + 15\), we have the equation:
\(CB\times CA=CD\times CE\)
Substituting the values, we get:
\(x(x + 21)=(x + 1)(x + 15)\)

Step2: Expand both sides of the equation

Expand the left - hand side: \(x(x + 21)=x^{2}+21x\)
Expand the right - hand side: \((x + 1)(x + 15)=x^{2}+15x+x + 15=x^{2}+16x + 15\)

Step3: Solve the resulting linear equation

Set the two expanded expressions equal to each other:
\(x^{2}+21x=x^{2}+16x + 15\)
Subtract \(x^{2}\) from both sides of the equation:
\(x^{2}-x^{2}+21x=x^{2}-x^{2}+16x + 15\)
\(21x=16x + 15\)
Subtract \(16x\) from both sides:
\(21x-16x=16x-16x + 15\)
\(5x=15\)
Divide both sides by 5:
\(x=\frac{15}{5}=3\)

Answer:

\(x = 3\)