Question

in the diagram, the length of the external portion of the secant segment pn is. the length of the entire secant segment ln is. the value of x is.

Explanation:

Step1: Identify the external portion of the secant segment

The external portion of the secant segment \(\overline{PN}\) is \(32\).

Step2: Calculate the length of the entire secant segment \(\overline{LN}\)

The length of the entire secant segment \(\overline{LN}\) is \(22 + 14=36\).

Step3: Use the secant - secant rule

By the secant - secant rule \(LN\times MN = PN\times QN\). Let \(QN=x\), \(LN = 36\), \(MN = 14\), \(PN=32 + x\). But we can also use the formula \( (external\ part\ of\ one\ secant)\times(whole\ secant)=(external\ part\ of\ another\ secant)\times(whole\ secant)\)

We know that for two secants \(LPN\) and \(MPN\) (where \(LN\) is one secant with external part \(MN = 14\) and whole length \(LN=36\), and \(PN\) is the other secant with external part \(32\) and whole length \(32 + x\)). The formula is \(14\times36=32\times(32 + x)\)

First, calculate \(14\times36 = 504\)

Then we have the equation \(504=32\times(32 + x)\)

Expand: \(504 = 1024+32x\)

Subtract \(1024\) from both sides: \(32x=504 - 1024=- 520\) (Wait, wrong formula. The correct formula is for two secants \(LN\) and \(PN\) (where \(LN\) has external part \(MN = 14\), whole length \(LM + MN=22 + 14 = 36\); \(PN\) has external part \(QN\) (let \(QN=x\), whole length \(x + 32\))

The correct formula is \(MN\times LN=QN\times PN\)

\(14\times(22 + 14)=x\times(x + 32)\)

\(14\times36=x^{2}+32x\)

\(504=x^{2}+32x\)

\(x^{2}+32x - 504 = 0\)

Factor: \(x^{2}+32x-504=(x + 42)(x - 12)=0\)

Since length cannot be negative, \(x = 12\)

Answer:

The length of the external portion of the secant segment \(\overline{PN}\) is \(32\). The length of the entire secant segment \(\overline{LN}\) is \(36\). The value of \(x\) is \(12\)