10 What are the coordinates of the point on the directed line segment from K(-5,-4) to L(5, 1) that partitions the segment into a ratio of 3 to 2?
10 What are the coordinates of the point on the directed line segment from K(-5,-4) to L(5, 1) that partitions the segment into a ratio of 3 to 2?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 74E
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![### Geometry Problem: Partitioning a Line Segment
#### Problem Statement:
What are the coordinates of the point on the directed line segment from \( K(-5, -4) \) to \( L(5, 1) \) that partitions the segment into a ratio of 3 to 2?
#### Detailed Explanation:
To solve this problem, we use the formula for finding a point that divides a line segment into a given ratio.
Given points:
- \( K(x_1, y_1) = (-5, -4) \)
- \( L(x_2, y_2) = (5, 1) \)
The ratio in which the segment is divided is 3:2.
Using the section formula for internal division:
\[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Here:
- \( m = 3 \)
- \( n = 2 \)
The coordinates of the point \( P \) are:
\[ P(x, y) = \left( \frac{3 \cdot 5 + 2 \cdot (-5)}{3+2}, \frac{3 \cdot 1 + 2 \cdot (-4)}{3+2} \right) \]
\[ P(x, y) = \left( \frac{15 - 10}{5}, \frac{3 - 8}{5} \right) \]
\[ P(x, y) = \left( \frac{5}{5}, \frac{-5}{5} \right) \]
\[ P(x, y) = (1, -1) \]
Therefore, the coordinates of the point that partitions the segment into a ratio of 3 to 2 are \( (1, -1) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb87196b7-c726-4ff8-b3f4-b62982cf63cc%2F906070b6-86c5-49a8-95ae-d7cbfbe9a040%2F80sk3mn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Partitioning a Line Segment
#### Problem Statement:
What are the coordinates of the point on the directed line segment from \( K(-5, -4) \) to \( L(5, 1) \) that partitions the segment into a ratio of 3 to 2?
#### Detailed Explanation:
To solve this problem, we use the formula for finding a point that divides a line segment into a given ratio.
Given points:
- \( K(x_1, y_1) = (-5, -4) \)
- \( L(x_2, y_2) = (5, 1) \)
The ratio in which the segment is divided is 3:2.
Using the section formula for internal division:
\[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Here:
- \( m = 3 \)
- \( n = 2 \)
The coordinates of the point \( P \) are:
\[ P(x, y) = \left( \frac{3 \cdot 5 + 2 \cdot (-5)}{3+2}, \frac{3 \cdot 1 + 2 \cdot (-4)}{3+2} \right) \]
\[ P(x, y) = \left( \frac{15 - 10}{5}, \frac{3 - 8}{5} \right) \]
\[ P(x, y) = \left( \frac{5}{5}, \frac{-5}{5} \right) \]
\[ P(x, y) = (1, -1) \]
Therefore, the coordinates of the point that partitions the segment into a ratio of 3 to 2 are \( (1, -1) \).
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