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What value of [tex]\( x \)[/tex] would make [tex]\( \overline{KM} \parallel \overline{JN} \)[/tex]?

By the converse of the side-splitter theorem, if [tex]\(\frac{JK}{KL} = \frac{x-5}{x}\)[/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]

Cross-multiply:
[tex]\[
(x-5)(x+4) = x(x-3)
\][/tex]

Distribute:
[tex]\[
x^2 + 4x - 5x - 20 = x^2 - 3x
\][/tex]

Combine like terms:
[tex]\[
x^2 - x - 20 = x^2 - 3x
\][/tex]

Subtract [tex]\( x^2 \)[/tex] from both sides:
[tex]\[
-x - 20 = -3x
\][/tex]

Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[
2x - 20 = 0
\][/tex]

Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 10
\][/tex]

Sagot :

GinnyAnswer
To determine the value of [tex]\( x \)[/tex] that makes [tex]\( \overline{ KM } \parallel \overline{ JN } \)[/tex] using the given proportion

[tex]\[ \frac{x-5}{x} = \frac{x-3}{x+4}, \][/tex]

we will solve this step-by-step.

### Step-by-Step Solution

1. Given Equation:

[tex]\[ \frac{x-5}{x} = \frac{x-3}{x+4} \][/tex]

2. Cross-multiply to clear the fractions:

[tex]\[ (x-5)(x+4) = x(x-3) \][/tex]

3. Distribute the terms:

- Distribute on the left side:
[tex]\[ x(x) + x(4) - 5(x) - 5(4) = x^2 + 4x - 5x - 20 \][/tex]
Simplifying this, we get:
[tex]\[ x^2 - x - 20 \][/tex]

- On the right side, distribute [tex]\( x \)[/tex] through [tex]\( (x-3) \)[/tex]:
[tex]\[ x(x) - x(3) = x^2 - 3x \][/tex]

4. Set the expressions equal to each other:

[tex]\[ x^2 - x - 20 = x^2 - 3x \][/tex]

5. Subtract [tex]\( x^2 \)[/tex] from both sides:

[tex]\[ -x - 20 = -3x \][/tex]

6. Isolate [tex]\( x \)[/tex]:

[tex]\[ -x + 3x = 20 \][/tex]
Simplify this to:
[tex]\[ 2x = 20 \][/tex]

7. Solve for [tex]\( x \)[/tex]:

[tex]\[ x = 10 \][/tex]

So, the value of [tex]\( x \)[/tex] that makes [tex]\( \overline{ KM } \parallel \overline{ JN } \)[/tex] is

[tex]\[ \boxed{10} \][/tex]
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