Question

which value of x would make \overline{lk} \parallel \overline{om}?
\\( x = 2 \\)
\\( x = 2.4 \\)
\\( x = 4.8 \\)
\\( x = 8 \\)

Explanation:

Step1: Apply the basic proportionality theorem (Thales' theorem)

If \( \overline{LK}\parallel\overline{OM}\), then \(\frac{NL}{LO}=\frac{NK}{KM}\).
Substitute \(NL = x\), \(LO=x - 4\), \(NK=x + 2\), and \(KM=x - 3\) into the proportion: \(\frac{x}{x - 4}=\frac{x + 2}{x - 3}\).

Step2: Cross - multiply

Cross - multiplying gives \(x(x - 3)=(x + 2)(x - 4)\).
Expand both sides:

• Left - hand side: \(x(x - 3)=x^{2}-3x\).
• Right - hand side: \((x + 2)(x - 4)=x^{2}-4x+2x - 8=x^{2}-2x - 8\).

So, \(x^{2}-3x=x^{2}-2x - 8\).

Step3: Solve for \(x\)

Subtract \(x^{2}\) from both sides of the equation \(x^{2}-3x=x^{2}-2x - 8\).
We get \(-3x=-2x - 8\).
Add \(2x\) to both sides: \(-3x+2x=-2x - 8+2x\).
\(-x=-8\).
Multiply both sides by \(- 1\) to get \(x = 8\).

Answer:

\(x = 8\) (the fourth option)