Question

applying the converse of the side-splitter theorem
which value of x would make \\(\overline{lk} \parallel \overline{om}\\)?
diagram: points n, k, m (top line: nk = x+2, km = x-3); points n, l, o (left line: nl = x, lo = x-4); segments lk and om
options:
\\(\circ\\) \\(x = 2\\)
\\(\circ\\) \\(x = 2.4\\)
\\(\circ\\) \\(x = 4.8\\)
\\(\circ\\) \\(x = 8\\)

Explanation:

Step1: Apply Side - Splitter Theorem

The Converse of the Side - Splitter Theorem states that if a line divides two sides of a triangle proportionally, then it is parallel to the third side. So, we have \(\frac{NL}{LO}=\frac{NK}{KM}\).
Given \(NL = x\), \(LO=x - 4\), \(NK=x + 2\), and \(KM=x - 3\). Substituting these values into the proportion, we get \(\frac{x}{x - 4}=\frac{x + 2}{x - 3}\).

Step2: Cross - Multiply

Cross - multiplying the proportion \(\frac{x}{x - 4}=\frac{x + 2}{x - 3}\) gives us \(x(x - 3)=(x + 2)(x - 4)\).

Step3: Expand Both Sides

Expanding the left - hand side: \(x(x - 3)=x^{2}-3x\).
Expanding the right - hand side: \((x + 2)(x - 4)=x^{2}-4x+2x - 8=x^{2}-2x - 8\).

Step4: Solve for x

Set the expanded forms equal to each other: \(x^{2}-3x=x^{2}-2x - 8\).
Subtract \(x^{2}\) from both sides: \(-3x=-2x - 8\).
Add \(2x\) to both sides: \(-3x + 2x=-2x+2x - 8\), which simplifies to \(-x=-8\).
Multiply both sides by \(- 1\): \(x = 8\).

Answer:

\(x = 8\) (corresponding to the option \(x = 8\))