Ratios of Directed Line Segments
Score: $4 / 6$ Penalty: 1 off
Question
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What are the coordinates of the point on the directed line segment from $(-7,-5)$ to $(8,-10)$ that partitions the segment into a ratio of 3 to 2 ?
Answer
Attempt 2 out of 2

Step 1 ：Use the formula for dividing a line segment in a given ratio: \( (x, y) = \left(\frac{x_1 \cdot m + x_2 \cdot n}{m + n}, \frac{y_1 \cdot m + y_2 \cdot n}{m + n}\right) \)

Step 2 ：Substitute the given values into the formula: \( (-7, -5) \) and \( (8, -10) \) are the endpoints, and the ratio is 3 to 2

Step 3 ：Calculate the x-coordinate: \( x = \frac{(-7) \cdot 3 + 8 \cdot 2}{3 + 2} = \frac{-21 + 16}{5} = \frac{-5}{5} = -1 \)

Step 4 ：Calculate the y-coordinate: \( y = \frac{(-5) \cdot 3 + (-10) \cdot 2}{3 + 2} = \frac{-15 - 20}{5} = \frac{-35}{5} = -7 \)

Step 5 ：Combine the x and y coordinates to get the final answer: \( \boxed{(-1, -7)} \)