what are the $x$- and $y$- coordinates of point p on the directed line segment from a to b such that p is $\frac{1}{3}$ the length of the line segment from a to b?

\x = \left(\frac{m}{m+n}\
ight)(x_2 - x_1) + x_1\
\y = \left(\frac{m}{m+n}\
ight)(y_2 - y_1) + y_1\

(1, 5)
(0, 3)
(-4, -5)
(-5, -7)

Question

what are the $x$- and $y$- coordinates of point p on the directed line segment from a to b such that p is $\frac{1}{3}$ the length of the line segment from a to b?

\x = \left(\frac{m}{m+n}\
ight)(x_2 - x_1) + x_1\
\y = \left(\frac{m}{m+n}\
ight)(y_2 - y_1) + y_1\

(1, 5)
(0, 3)
(-4, -5)
(-5, -7)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Directed Line Segments", "Section Formula" ], "new_concepts": [], "current_concepts": [ "Directed Line Segments", "Section Formula", "Midpoint Partition Ratio" ] } ## Identify coordinates and partition ratio $$ \begin{aligned} &A(x_1, y_1) = (-8, -13)\ &B(x_2, y_2) = (4, 11)\ &\frac{m}{m+n} = \frac{1}{3} \end{aligned} $$ ## Calculate the x-coordinate of point P $$ \begin{aligned} x &= \frac{1}{3}(4 - (-8)) + (-8)\ x &= \frac{1}{3}(12) - 8\ x &= 4 - 8 = -4 \end{aligned} $$ ## Calculate the y-coordinate of point P $$ \begin{aligned} y &= \frac{1}{3}(11 - (-13)) + (-13)\ y &= \frac{1}{3}(24) - 13\ y &= 8 - 13 = -5 \end{aligned} $$

(A) $(1, 5)$

(B) $(0, 3)$

(C) $(-4, -5)$

(D) $(-5, -7)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Directed Line Segments" ] }

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QUESTION IMAGE

Question

Explanation:

Identify coordinates and partition ratio

Calculate the x-coordinate of point P

Calculate the y-coordinate of point P

Answer: