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What value of $x$ would make $KM \parallel JN$?



Complete the statements to solve for $x$.

By the converse of the side-splitter theorem, if $\frac{JK}{KL} =$ [ ] then $\overline{ KM } \parallel \overline{ JN }$.

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 $x$: $x =$ [ ].

Verified Answer :

  • The proportionality condition for parallel lines, \(\frac{JK}{KL} = \frac{NM}{ML}\), leads to the equation \(\frac{x-5}{x} = \frac{x-3}{x+4}\).
  • Cross-multiplying results in \((x-5)(x+4) = x(x-3)\).
  • Expanding the sides gives \(x^2 - x - 20 = x^2 - 3x\).
  • Simplifying this linear equation leads to the solution \(\boxed{x=10}\).

Explanation

  1. Analyze the proportionality condition.

The converse of the side-splitter theorem states that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side. In the context of two transversals intersected by lines, for \(\overline{KM} \parallel \overline{JN}\), the ratio of the segments on the transversals must be equal: \(\frac{JK}{KL} = \frac{NM}{ML}\). We are given the proportion \(\frac{x-5}{x} = \frac{x-3}{x+4}\). To find the value of \(x\) that makes the lines parallel, we solve this proportionality equation.

  1. Expand and equate the expressions.

To solve for \(x\), we cross-multiply the equation:
\[ (x-5)(x+4) = x(x-3) \]
Expanding the left side:
\[ (x \cdot x) + (x \cdot 4) - (5 \cdot x) - (5 \cdot 4) = x^2 + 4x - 5x - 20 = x^2 - x - 20 \]
Expanding the right side:
\[ x \cdot x - x \cdot 3 = x^2 - 3x \]
Setting them equal gives:
\[ x^2 - x - 20 = x^2 - 3x \]

  1. Solve the linear equation for \(x\).

Now, subtract \(x^2\) from both sides of the equation to simplify:
\[ -x - 20 = -3x \]
Add \(3x\) to both sides:
\[ 2x - 20 = 0 \]
Add \(20\) to both sides:
\[ 2x = 20 \]
Dividing by 2 gives:
\[ x = 10 \]
Substituting \(x=10\) back into the original expressions: \(\frac{10-5}{10} = \frac{5}{10} = 0.5\) and \(\frac{10-3}{10+4} = \frac{7}{14} = 0.5\). The equality holds. Therefore, the value of \(x\) that satisfies the condition is \(10\).

Examples

This concept of proportional segments is used in engineering and construction for creating parallel components. For instance, when designing a staircase or structural braces, engineers use the side-splitter theorem (or its converse) to ensure that support bars or railings are perfectly parallel by checking that the ratios of the lengths of the segments created on the support beams are equal. If \(\frac{JK}{KL} = \frac{NM}{ML}\), they can be confident that the pieces are aligned correctly.

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