To find the value of [tex]\( x \)[/tex] that would make [tex]\(\overline{ KM } \parallel \overline{ JN }\)[/tex], we need to use the given proportion and solve for [tex]\( x \)[/tex]. Here is the detailed step-by-step solution:
1. Identify the proportion:
By the converse of the side-splitter theorem, if [tex]\(\frac{JK}{KL} = \frac{KM}{MN}\)[/tex], then [tex]\(\overline{ KM } \parallel \overline{ JN }\)[/tex].
According to the problem:
[tex]\[
\frac{x-5}{x} = \frac{x-3}{x+4}
\][/tex]
2. Cross-multiply the given proportion:
Cross-multiplying the fractions, we get:
[tex]\[
(x - 5)(x + 4) = x(x - 3)
\][/tex]
3. Distribute and simplify both sides:
First, expand both sides:
[tex]\[
(x - 5)(x + 4) = x^2 + 4x - 5x - 20 = x^2 - x - 20
\][/tex]
[tex]\[
x(x - 3) = x^2 - 3x
\][/tex]
4. Set the expressions equal to each other and solve for [tex]\( x \)[/tex]:
Equating the two expanded expressions, we have:
[tex]\[
x^2 - x - 20 = x^2 - 3x
\][/tex]
5. Move all terms to one side to simplify the equation:
Subtract [tex]\( x^2 \)[/tex] from both sides:
[tex]\[
-x - 20 = -3x
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[
2x - 20 = 0
\][/tex]
Now isolate [tex]\( x \)[/tex] by adding 20 to both sides and then dividing by 2:
[tex]\[
2x = 20
\][/tex]
[tex]\[
x = \frac{20}{2}
\][/tex]
[tex]\[
x = 10
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that makes [tex]\(\overline{ KM } \parallel \overline{ JN }\)[/tex] is:
[tex]\[
x = 10
\][/tex]
