Question

point p partitions the directed line segment from a to b into the ratio 3 : 4. will p be closer to a or b? why? image of a line segment with points a and b four multiple-choice options: p will be closer to b because it will be 4/7 the distance from b to a; p will be closer to a because it will be 4/7 the distance from a to b; p will be closer to a because it will be 3/7 the distance from a to b; p will be closer to b because it will be 3/7 the distance from b to a

Explanation:

Step1: Understand the Ratio

The ratio of \( AP:PB = 3:4 \), so the total parts of \( AB \) is \( 3 + 4 = 7 \) parts.

Step2: Analyze Distance from A

The length from \( A \) to \( P \) is \( 3 \) parts, and from \( A \) to \( B \) is \( 7 \) parts. So \( AP=\frac{3}{7}AB \), and \( PB = \frac{4}{7}AB \). Since \( \frac{3}{7}<\frac{4}{7} \), \( P \) is closer to \( A \) as \( AP \) is a smaller fraction of \( AB \). Now check the options: the third option says "P will be closer to A because it will be \( \frac{3}{7} \) the distance from A to B" which matches.

Answer:

P will be closer to A because it will be \( \frac{3}{7} \) the distance from A to B. (The third option among the given choices)