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?

p will be closer to a because it will be \\(\frac{3}{7}\\) the distance from a to b.
p will be closer to a because it will be \\(\frac{4}{7}\\) the distance from a to b.
p will be closer to b because it will be \\(\frac{3}{7}\\) the distance from b to a.
p will be closer to b because it will be \\(\frac{4}{7}\\) the distance from b to a.

Explanation:

Analyze the partition ratio

$$ \text{Ratio } AP : PB = 3 : 4 $$

Calculate fractional distance from A to B

$$ \frac{AP}{AB} = \frac{3}{3 + 4} = \frac{3}{7} $$

Determine proximity to endpoints

$$ \frac{3}{7} < \frac{1}{2} \implies AP < PB \implies P \text{ is closer to } A $$

Answer:

• P will be closer to A because it will be \(\frac{3}{7}\) the distance from A to B. (Correct answer)
• P will be closer to A because it will be \(\frac{4}{7}\) the distance from A to B.
• P will be closer to B because it will be \(\frac{3}{7}\) the distance from B to A.
• P will be closer to B because it will be \(\frac{4}{7}\) the distance from B to A.