Question

which statement is true about the diagram?
a---k---b---t (with a to b labeled 10, k to b labeled 5, b to t labeled 4)
options:

• k is the midpoint of \\(\overline{ab}\\).
• b is the midpoint of \\(\overline{kt}\\).
• \\(ak = bt\\)
• \\(ab = kt\\)

Explanation:

1. Analyze each option:

• Option 1: For \( K \) to be the midpoint of \( \overline{AB} \), \( AK = KB \). Given \( AB = 10 \), if \( K \) is the midpoint, \( AK = KB = 5 \). From the diagram, \( KB = 5 \), and \( AK=AB - KB=10 - 5 = 5 \), so \( AK = KB = 5 \), but wait, let's check other options too.
• Option 2: For \( B \) to be the midpoint of \( \overline{KT} \), \( KB=BT \). \( KB = 5 \), \( BT = 4 \), so \( 5

eq4 \), so this is false.

• Option 3: \( AK = AB - KB=10 - 5 = 5 \), \( BT = 4 \), \( 5

eq4 \), so this is false.

• Option 4: \( AB = 10 \), \( KT=KB + BT=5 + 4 = 9 \), \( 10

eq9 \), so this is false. Wait, earlier calculation for option 1: Wait, the length of \( AB \) is 10. If \( K \) is a point on \( AB \) such that \( KB = 5 \), then \( AK=AB - KB = 10 - 5=5 \). So \( AK = KB = 5 \), which means \( K \) is the midpoint of \( \overline{AB} \). Wait, maybe I made a mistake in option 4. Wait \( KT=KB + BT = 5 + 4=9 \), \( AB = 10 \), so \( AB
eq KT \). Option 1: \( AK = 5 \), \( KB = 5 \), so \( K \) is the midpoint of \( \overline{AB} \). Let's re - check:

• Length of \( AB = 10 \). The distance from \( A \) to \( K \): since \( AB = 10 \) and \( KB = 5 \), then \( AK=AB - KB = 10 - 5 = 5 \). So \( AK = KB = 5 \), so \( K \) divides \( AB \) into two equal parts. So \( K \) is the midpoint of \( \overline{AB} \).

Answer:

\( K \) is the midpoint of \( \overline{AB} \)