Question

in the diagram, line a is the perpendicular bisector of \\(\overline{km}\\). what is the length of \\(\overline{km}\\)? \\(\bigcirc\\) 22 units \\(\bigcirc\\) 40 units \\(\bigcirc\\) 44 units \\(\bigcirc\\) 80 units (diagram: triangle with points k, n, m; line a with point n, segment kl: 6x + 4, kn: 9x - 5, nm: 7x + 7, l on km, n on line a)

Explanation:

Step1: Use perpendicular bisector property

Since line \( a \) is the perpendicular bisector of \( \overline{KM} \), \( NK = NM \) and \( KL = LM \). First, use \( NK = NM \) to find \( x \).
\( 9x - 5 = 7x + 7 \)

Step2: Solve for \( x \)

Subtract \( 7x \) from both sides: \( 2x - 5 = 7 \)
Add 5 to both sides: \( 2x = 12 \)
Divide by 2: \( x = 6 \)

Step3: Find length of \( KL \)

Substitute \( x = 6 \) into \( KL = 6x + 4 \): \( KL = 6(6) + 4 = 36 + 4 = 40 \)

Step4: Find length of \( KM \)

Since \( KL = LM \), \( KM = KL + LM = 2 \times KL \). So \( KM = 2 \times 40 = 80 \)? Wait, no, wait. Wait, \( KL = 6x + 4 = 40 \)? Wait, no, wait, \( x = 6 \), so \( 6x + 4 = 6*6 +4 = 40 \). Then \( KM = KL + LM = 40 + 40 = 80 \)? But wait, let's check again. Wait, maybe I made a mistake. Wait, \( NK = 9x -5 = 9*6 -5 = 54 -5 = 49 \), \( NM = 7x +7 = 7*6 +7 = 49 \), so that's correct. Then \( KL = 6x +4 = 40 \), so \( LM = 40 \), so \( KM = 40 + 40 = 80 \). Wait, but the options include 80. Wait, but let me check again. Wait, maybe the question is different. Wait, no, the diagram: line \( a \) is perpendicular bisector, so \( L \) is the midpoint, so \( KL = LM \), and \( N \) is on the perpendicular bisector, so \( NK = NM \). So we found \( x = 6 \), \( KL = 40 \), so \( KM = 2*40 = 80 \). Wait, but let's check the options. The options are 22, 40, 44, 80. Wait, but maybe I messed up. Wait, no, \( KL = 6x +4 = 6*6 +4 = 40 \), so \( KM = KL + LM = 40 + 40 = 80 \). So the length of \( KM \) is 80 units.

Answer:

80 units