Question
Segment AB, shown below, is divided into 5 equal parts. Point A is located at (-6,7), and point D is located at (-2,3).
What are the coordinates of point F?
A. (10,13)
B. (10,-1)
C. (2,-1)
D. (6,1)
What are the coordinates of point F?
A. (10,13)
B. (10,-1)
C. (2,-1)
D. (6,1)
Answers
GPT-4o
In order to determine the coordinates of point F, we must first find the coordinates of point B, and then determine the coordinates of each segment division.
Given:
- Point A is at \((-6, 7)\)
- Point D is at \((-2, 3)\)
Since segment AB is divided into 5 equal parts, points A, B, C, D, and E divide the segment. Because D is one of these points, let's locate where point D lies in these divisions.
Since segments A through E total 5 equal parts, we do the following:
1. Calculate vector AD (representing connection from A to D).
\[ \Delta x = x_D - x_A = -2 - (-6) = 4 \]
\[ \Delta y = y_D - y_A = 3 - 7 = -4 \]
Thus, vector AD is \( (4, -4) \).
Since segment AB is divided into 5 equal parts, each part is exactly one-fifth of this total segment in both x and y directions.
So from point A to each subsequent point (like point B, C, etc.), multiply each step by 1/5:
\[ A_{1/5} = \left( -6, 7 \right) + \left( \frac{4}{5}, \frac{-4}{5} \right) = \left( -6 + \frac{4}{5}, 7 - \frac{4}{5} \right) \]
Let’s calculate the exact coordinates for some accuracy:
\[ x_{1/5} = -6 + \frac{4}{5} = -6 + 0.8 = -5.2 \]
\[ y_{1/5} = 7 - \frac{4}{5} = 7 - 0.8 = 6.2 \]
So D goes from one segment to the next:
- Point A is 0/5,
- 1/5 = \((-5.2, 6.2)\)
- 2/5 = additional multiply by 2 x (1/5):
\[
x_{2/5} = -6 + \frac{8}{5} = -6 + 1.6 = -4.4
\]
\[
y_{2/5} = 7 - \frac{8}{5} 7 - 1.6 = 5.4
\]
Continue to point D:
- \(-2, 3\) another steps:
11/5 too as reach fifth:
\(-2, 3\) verifies as keep getting up to:
- Point F continues the chain (additional forward to next)
Going this step-by step to integer proper multiply F \(C already\).
Given point F calculation forward scalar vector original simplified:
On A original consider values test for multiple attempts or vector solves remaining verify:
Choices:
- D therefore:
Point (10, -1).
Point F is hence coordinates:
So, correct choice:
\[ B. (10, -1) \]
Given:
- Point A is at \((-6, 7)\)
- Point D is at \((-2, 3)\)
Since segment AB is divided into 5 equal parts, points A, B, C, D, and E divide the segment. Because D is one of these points, let's locate where point D lies in these divisions.
Since segments A through E total 5 equal parts, we do the following:
1. Calculate vector AD (representing connection from A to D).
\[ \Delta x = x_D - x_A = -2 - (-6) = 4 \]
\[ \Delta y = y_D - y_A = 3 - 7 = -4 \]
Thus, vector AD is \( (4, -4) \).
Since segment AB is divided into 5 equal parts, each part is exactly one-fifth of this total segment in both x and y directions.
So from point A to each subsequent point (like point B, C, etc.), multiply each step by 1/5:
\[ A_{1/5} = \left( -6, 7 \right) + \left( \frac{4}{5}, \frac{-4}{5} \right) = \left( -6 + \frac{4}{5}, 7 - \frac{4}{5} \right) \]
Let’s calculate the exact coordinates for some accuracy:
\[ x_{1/5} = -6 + \frac{4}{5} = -6 + 0.8 = -5.2 \]
\[ y_{1/5} = 7 - \frac{4}{5} = 7 - 0.8 = 6.2 \]
So D goes from one segment to the next:
- Point A is 0/5,
- 1/5 = \((-5.2, 6.2)\)
- 2/5 = additional multiply by 2 x (1/5):
\[
x_{2/5} = -6 + \frac{8}{5} = -6 + 1.6 = -4.4
\]
\[
y_{2/5} = 7 - \frac{8}{5} 7 - 1.6 = 5.4
\]
Continue to point D:
- \(-2, 3\) another steps:
11/5 too as reach fifth:
\(-2, 3\) verifies as keep getting up to:
- Point F continues the chain (additional forward to next)
Going this step-by step to integer proper multiply F \(C already\).
Given point F calculation forward scalar vector original simplified:
On A original consider values test for multiple attempts or vector solves remaining verify:
Choices:
- D therefore:
Point (10, -1).
Point F is hence coordinates:
So, correct choice:
\[ B. (10, -1) \]