To determine the value of [tex]\(x\)[/tex] that would make [tex]\(\overline{KM} \parallel \overline{JN}\)[/tex], we can use the converse of the Side-Splitter Theorem. Here are the steps:
1. Identify the given ratio: By the converse of the Side-Splitter Theorem, if [tex]\( \frac{JK}{KL} = \frac{x-5}{4} \)[/tex] then [tex]\( \overline{KM} \parallel \overline{JN} \)[/tex].
2. Form the proportion: According to the theorem, we use the proportion [tex]\( \frac{x-5}{4} = \frac{3}{x} \)[/tex].
3. Cross-multiply to form an equation: [tex]\((x-5) \cdot x = 4 \cdot 3\)[/tex].
4. Distribute and set up the equation:
[tex]\[
x(x) + x(-5) = 4 \cdot 3.
\][/tex]
Simplifying, we get:
[tex]\[
x^2 - 5x = 12.
\][/tex]
5. Move all terms to one side of the equation to set it to zero:
[tex]\[
x^2 - 5x - 12 = 0.
\][/tex]
6. Solve the quadratic equation for [tex]\(x\)[/tex]:
The solutions to the quadratic equation [tex]\(x^2 - 5x - 12 = 0\)[/tex] can be found using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = -12\)[/tex].
By solving the quadratic equation, we get the values of [tex]\(x\)[/tex] as:
[tex]\[
x = \frac{5}{2} - \frac{\sqrt{73}}{2} \quad \text{or} \quad x = \frac{5}{2} + \frac{\sqrt{73}}{2}.
\][/tex]
Therefore, the two values of [tex]\(x\)[/tex] that satisfy the condition are:
[tex]\[
x = \frac{5}{2} - \frac{\sqrt{73}}{2} \quad \text{and} \quad x = \frac{5}{2} + \frac{\sqrt{73}}{2}.
\][/tex]
