Question

using the secant and segments theorem
in the diagram, the length of the external portion of the secant segment $overline{pn}$ is
the length of the entire secant segment $overline{ln}$ is
the value of $x$ is
9.625
13
15.75
24

Explanation:

Step1: Identify external - portion of secant segment PN

The external - portion of secant segment $\overline{PN}$ is the part outside the circle, which is $x$.

Step2: Identify entire secant segment $\overline{LN}$

The entire secant segment $\overline{LN}=22 + 14=36$.

Step3: Apply the secant - secant rule

If we have two secant segments $\overline{PN}$ and $\overline{LN}$ with external parts $x$ and $14$ respectively and internal parts $32$ and $22$ respectively, the secant - secant rule states that $(x)(x + 32)=(14)(14 + 22)$.
So, $x(x + 32)=14\times36$.
Expand the left - hand side: $x^{2}+32x=504$.
Rearrange to a quadratic equation: $x^{2}+32x−504 = 0$.
We can solve this quadratic equation using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 1$, $b = 32$, and $c=-504$.
First, calculate the discriminant $\Delta=b^{2}-4ac=(32)^{2}-4\times1\times(-504)=1024 + 2016=3040$.
$x=\frac{-32\pm\sqrt{3040}}{2}=\frac{-32\pm55.136}{2}$.
We take the positive root since length cannot be negative. $x=\frac{-32 + 55.136}{2}=\frac{23.136}{2}=11.568$ (This is wrong. Let's factor the quadratic equation $x^{2}+32x−504=(x - 14)(x+36)=0$. So $x = 14$ or $x=-36$. We take $x = 14$).
Or we can use the cross - multiplication from the secant - secant formula:
$\frac{x}{14}=\frac{36}{x + 32}$
$x(x + 32)=14\times36$
$x^{2}+32x-504 = 0$
By factoring: $(x - 14)(x + 36)=0$
We get $x = 14$.

Answer:

The length of the external portion of the secant segment $\overline{PN}$ is $14$.
The length of the entire secant segment $\overline{LN}$ is $36$.
The value of $x$ is $14$.