Question

what is the value of x? 2 3 6 7

Explanation:

Step1: Apply secant - secant rule

If two secants are drawn to a circle from an exterior point, then $(EA)\times(EB)=(ED)\times(EC)$. Here, $EA = 11+(x + 1)=x + 12$, $EB=x + 1$, $ED = 1+(x + 4)=x+5$, $EC = 1$. So, $(x + 12)(x + 1)=(x + 5)\times1$.

Step2: Expand left - hand side

Expand $(x + 12)(x + 1)$ using FOIL method: $x^{2}+x+12x + 12=x^{2}+13x + 12$. The equation becomes $x^{2}+13x + 12=x + 5$.

Step3: Rearrange to quadratic form

Subtract $x$ and $5$ from both sides: $x^{2}+13x - x+12 - 5=0$, which simplifies to $x^{2}+12x + 7 = 0$. This is wrong. Let's use the correct secant - secant rule: $(EB)\times(EA)=(EC)\times(ED)$. So, $(x + 1)(x + 1+11)=(x + 4)(x + 4 + 1)$.
$(x + 1)(x+12)=(x + 4)(x + 5)$.

Step4: Expand both sides

Expand: $x^{2}+12x+x + 12=x^{2}+5x+4x + 20$.
$x^{2}+13x + 12=x^{2}+9x + 20$.

Step5: Solve for x

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: $4x=20 - 12=8$. Divide both sides by 4: $x = 2$.

Answer:

A. 2