Question

applying the converse of the side-splitter theorem
which value of x would make \\(\overline{lk} \parallel \overline{om}\\)?
\\(\bigcirc\\ x=2\\)
\\(\bigcirc\\ x=2.4\\)
\\(\bigcirc\\ x=4.8\\)
\\(\bigcirc\\ x=8\\)
(image shows a geometric figure with points n, k, m on a line segment, and points n, l, o on another line segment, with lengths labeled: nk = x + 2, km = x - 3, nl = x, lo = x - 4, and lk and om as line segments.)

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. In triangle \( \triangle ONM \), line \( LK \) divides \( ON \) and \( NM \) proportionally. So, we have the proportion \(\frac{OL}{LN}=\frac{MK}{KN}\).

We know that \( OL=x - 4\), \( LN=x\), \( MK=x - 3\), and \( KN=x + 2\). Substituting these values into the proportion, we get \(\frac{x-4}{x}=\frac{x - 3}{x + 2}\).

Step2: Cross - Multiply

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

Expanding both sides:

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

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

Step3: Solve for \(x\)

Subtract \(x^{2}\) from both sides of the equation \(x^{2}-2x - 8=x^{2}-3x\):

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

Simplifying, we get \(-2x - 8=-3x\).

Add \(3x\) to both sides: \(-2x+3x - 8=-3x + 3x\), which simplifies to \(x - 8 = 0\).

Add 8 to both sides: \(x=8\).

Answer:

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