What value of [tex]\( x \)[/tex] would make [tex]\( \overline{KM} \parallel \overline{JN} \)[/tex]?
Complete the statements to solve for [tex]\( x \)[/tex].
By the converse of the side-splitter theorem, if [tex]\( \frac{JK}{KL} = \square \)[/tex], then [tex]\( \overline{KM} \parallel \overline{JN} \)[/tex].
Substitute the expressions into the proportion: [tex]\[
\frac{x-5}{x} = \frac{x-3}{x+4}
\][/tex]
To find the value of [tex]\(x\)[/tex] that would make [tex]\(\overline{KM} \parallel \overline{JN}\)[/tex], we can use the side-splitter theorem. According to the theorem, if [tex]\(\overline{KM}\)[/tex] is parallel to [tex]\(\overline{JN}\)[/tex], then the ratio [tex]\( \frac{JK}{KL} \)[/tex] must be equal to the ratio [tex]\( \frac{KM}{MN} \)[/tex].
Given the problem: [tex]\[ \frac{x-5}{x}=\frac{x-3}{x+4} \][/tex]
Follow these steps to solve for [tex]\(x\)[/tex]:
1. Cross-multiply the given proportion: [tex]\[ (x-5)(x+4) = x(x-3) \][/tex]
3. Simplify the equation: [tex]\[ x^2 - x - 20 = x^2 - 3x \][/tex]
4. To eliminate [tex]\(x^2\)[/tex] from both sides, subtract [tex]\(x^2\)[/tex] from both sides of the equation: [tex]\[ -x - 20 = -3x \][/tex]
5. Solve for [tex]\(x\)[/tex]: [tex]\[ -x - 20 = -3x \][/tex] [tex]\[ -20 = -3x + x \][/tex] [tex]\[ -20 = -2x \][/tex] [tex]\[ x = \frac{20}{2} \][/tex] [tex]\[ x = 10 \][/tex]
So, the value of [tex]\(x\)[/tex] that makes [tex]\(\overline{KM} \parallel \overline{JN}\)[/tex] is [tex]\(x = 10\)[/tex]:
Summarizing the steps with the final solution: - By the converse of the side-splitter theorem, if [tex]\( \frac{JK}{KL} = \frac{KM}{MN} \)[/tex]: [tex]\[ \frac{x-5}{x} = \frac{x-3}{x+4} \][/tex]
- Substitute the expressions into the proportion: [tex]\[ \frac{x-5}{x} = \frac{x-3}{x+4} \][/tex]
