Question

which statement is true about the diagram?
diagram of a line segment with points a, k, b, t. the length from a to k to b is labeled 10 (a to k is part, k to b is 5), and from b to t is 4. below the diagram are four options: ab = kt; b is the midpoint of \overline{kt}; ak = bt; k is the midpoint of \overline{ab}.

Explanation:

Step1: Analyze each option

• Option 1: \( AB = 10 \), \( KT = 5 + 4 = 9 \), so \( AB

eq KT \).

• Option 2: \( B \) is the midpoint of \( \overline{KT} \)? \( KT = 9 \), midpoint would be at \( 4.5 \) from \( K \), but \( KB = 5 \), \( BT = 4 \), so no.
• Option 3: \( AK \): From \( A \) to \( K \), \( AB = 10 \), \( KB = 5 \), so \( AK = AB - KB = 10 - 5 = 5 \). \( BT = 4 \)? Wait, no, wait. Wait, \( AK \): Let's re - calculate. The length from \( A \) to \( K \): The total length from \( A \) to \( B \) is 10, from \( K \) to \( B \) is 5, so \( AK = 10 - 5 = 5 \). The length from \( B \) to \( T \) is 4? Wait, no, \( KT \) is \( 5 + 4 = 9 \), \( AB = 10 \). Wait, \( AK \): \( A \) to \( K \): the segment from \( A \) to \( K \): since \( A \) to \( B \) is 10, and \( K \) is between \( A \) and \( B \) (wait, no, the diagram: \( A---K---B---T \)). So \( AK \): length from \( A \) to \( K \), \( KB \) is 5, \( AB \) is 10, so \( AK = AB - KB = 10 - 5 = 5 \). \( BT \): length from \( B \) to \( T \) is 4? Wait, no, \( KT \) is \( KB + BT = 5 + 4 = 9 \), \( AB = 10 \). Wait, \( AK \): \( A \) to \( K \): let's see the positions. \( A \) is at the top, \( K \) is below \( A \), \( B \) below \( K \), \( T \) below \( B \). So \( AK \): the length from \( A \) to \( K \): if \( AB = 10 \) and \( KB = 5 \), then \( AK = 10 - 5 = 5 \). \( BT = 4 \)? No, wait, \( BT \) is 4? Wait, no, \( KT = 5 + 4 = 9 \), \( AB = 10 \). Wait, maybe I misread. Wait, the diagram: \( A \) to \( K \) to \( B \) to \( T \). So \( AK \): distance from \( A \) to \( K \), \( KB \) is 5, \( AB \) is 10, so \( AK = 10 - 5 = 5 \). \( BT \): distance from \( B \) to \( T \) is 4? No, \( KT = KB + BT = 5 + 4 = 9 \), \( AB = 10 \). Wait, no, maybe \( AK \) is 5, and \( BT \) is 4? No, that can't be. Wait, maybe I made a mistake. Wait, let's check the fourth option: \( K \) is the midpoint of \( \overline{AB} \)? \( AB = 10 \), midpoint would be at 5 from \( A \), and \( AK = 5 \) (since \( AB = 10 \), \( AK = 5 \)), so \( K \) is the midpoint of \( \overline{AB} \) because \( AK = KB = 5 \) (since \( KB = 5 \), \( AK = 5 \)). Oh! I see, I misread the positions. \( A---K---B \), so \( AK = 5 \), \( KB = 5 \), so \( K \) is the midpoint of \( \overline{AB} \) (since \( AB = AK + KB = 5 + 5 = 10 \), so midpoint is at 5 from \( A \), which is \( K \)).

• Option 4: \( K \) is the midpoint of \( \overline{AB} \)? \( AB = 10 \), \( AK = 5 \), \( KB = 5 \), so yes, \( K \) is the midpoint of \( \overline{AB} \). Wait, but let's re - check the options. The options are:

1. \( AB = KT \)
2. \( B \) is the midpoint of \( \overline{KT} \)
3. \( AK = BT \)
4. \( K \) is the midpoint of \( \overline{AB} \)

Wait, my initial analysis was wrong. Let's re - do:

• Option 1: \( AB = 10 \), \( KT = 5 + 4 = 9 \), so \( AB

eq KT \).

• Option 2: \( KT = 9 \), midpoint would be at \( 4.5 \) from \( K \), \( KB = 5 \), \( BT = 4 \), so \( B \) is not the midpoint.
• Option 3: \( AK \): \( A \) to \( K \): since \( AB = 10 \), \( KB = 5 \), \( AK = AB - KB = 10 - 5 = 5 \). \( BT = 4 \), so \( AK

eq BT \). Wait, I was wrong earlier.

• Option 4: \( K \) is the midpoint of \( \overline{AB} \): \( AB = 10 \), \( AK = 5 \), \( KB = 5 \), so \( AK = KB \), so \( K \) is the midpoint of \( \overline{AB} \).

Step2: Confirm the correct option

After re - analyzing, the correct option is the one where \( K \) is the midpoint of \( \overline{AB} \) (the fourth option). Wait, but in the original options, the fourth option is "K is the midpoint of \( \overline{AB} \)".

Answer:

The statement "K is the midpoint of \( \overline{AB} \)" is true. (Assuming the options are presented as: 1. \( AB = KT \); 2. \( B \) is the midpoint of \( \overline{KT} \); 3. \( AK = BT \); 4. \( K \) is the midpoint of \( \overline{AB} \), the correct option is the fourth one: "K is the midpoint of \( \overline{AB} \)")