Question

what is the value of x? 3 7 6 2

Explanation:

Step1: Apply the secant - secant rule

If two secant segments are drawn to a circle from an external point, then \((EA)(EB)=(ED)(EC)\). Here, \(EA = 11+(x + 1)=x+12\), \(EB=x + 1\), \(ED=1+(x + 4)=x + 5\), \(EC=x + 4\). But another way (using the formula for the product of the lengths of the secant segments): \((\text{whole secant})\times(\text{external part})=(\text{whole secant})\times(\text{external part})\). The formula is \((EB + BA)\times EB=(EC+CD)\times EC\). Substituting the values: \((x + 1+11)\times(x + 1)=(x + 4+1)\times(x + 4)\)

Step2: Expand the equation

Expand \((x + 12)(x + 1)\) and \((x + 5)(x + 4)\)

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Step3: Solve the linear equation

Set \(x^{2}+13x+12=x^{2}+9x + 20\). Subtract \(x^{2}\) from both sides: \(13x+12=9x + 20\). Subtract \(9x\) from both sides: \(13x-9x+12=9x-9x + 20\), \(4x+12 = 20\). Subtract 12 from both sides: \(4x=20 - 12\), \(4x=8\). Divide both sides by 4: \(x = 2\)

Answer:

\(2\)