Question

applying the converse of the side - splitter theorem
what value of x would make \\( \overline { k m } \parallel \overline { j n } \\)?
complete the statements to solve for x.
by the converse of the side - splitter theorem, if \\( j k / k l = \\)
, then \\( \overline { k m } \parallel \overline { j n } \\).
substitute the expressions into the proportion:
\\( \frac { x - 5 } { x } = \frac { x - 3 } { x + 4 } \\).
cross - multiply: \\( ( x - 5 ) ( ) = x ( x - 3 ) \\).
distribute: \\( x ( x ) + x ( 4 ) - 5 ( x ) - 5 ( 4 ) = x ( x ) + x ( - 3 ) \\).
multiply and simplify: \\( x ^ { 2 } - x - = x ^ { 2 } - 3 x \\).
solve for x: \\( x = \\)

Explanation:

Step1: Fill in the first blank

By the converse of the side - splitter theorem, if \(\frac{JK}{KL}=\frac{JM}{MN}\), then \(\overline{KM}\parallel\overline{JN}\).

Step2: Cross - multiply

Given \(\frac{x - 5}{x}=\frac{x - 3}{x + 4}\), cross - multiply: \((x - 5)(x + 4)=x(x - 3)\)

Step3: Distribute and simplify

$$
LATEXBLOCK0
$$

Step4: Solve for \(x\)

Subtract \(x^{2}\) from both sides: \(-x-20=-3x\)
Add \(3x\) to both sides: \(2x-20 = 0\)
Add \(20\) to both sides: \(2x=20\)
Divide both sides by \(2\): \(x = 10\)

Answer:

First blank: \(\frac{JM}{MN}\), second blank: \(x + 4\), third blank: \(20\), fourth blank: \(10\)