Question

which value of x would make \\(\overline{no} \parallel \overline{kj}\\)?
options: 8, 1, 6, 10
figure: triangle lkj with point n on lk and point o on lj. ln = (x - 3) in, nk = (x + 2) in, lo = (x - 4) in, oj = x in.

Explanation:

Step1: Apply the Basic Proportionality Theorem (Thales' theorem)

If \(\overline{NO} \parallel \overline{KJ}\), then by the Basic Proportionality Theorem (Thales' theorem), \(\frac{LN}{NK}=\frac{LO}{OJ}\).
We know \(LN=(x - 3)\) in, \(NK=(x + 2)\) in, \(LO=(x - 4)\) in, and \(OJ = x\) in. So the proportion is \(\frac{x - 3}{x + 2}=\frac{x - 4}{x}\).

Step2: Cross - multiply to solve the proportion

Cross - multiplying gives \(x(x - 3)=(x - 4)(x + 2)\).
Expanding both sides:
Left side: \(x^{2}-3x\)
Right side: \(x^{2}+2x-4x - 8=x^{2}-2x - 8\)
So we have the equation \(x^{2}-3x=x^{2}-2x - 8\).

Step3: Solve for \(x\)

Subtract \(x^{2}\) from both sides: \(-3x=-2x - 8\)
Add \(2x\) to both sides: \(-3x + 2x=-8\)
Simplify: \(-x=-8\)
Multiply both sides by \(- 1\): \(x = 8\)

Answer:

8