Given that point D divides the line segment AB in the ratio of 5:3, we need to determine the coordinates of point C on the vertical line segment AC where CD is horizontal. The problem states that AC is vertical and CD is horizontal and lists multiple coordinate options for point C.
Let's denote the coordinates of points A, B, C, and D as follows:
- \( A = (x_1, y_1) \)
- \( B = (x_2, y_2) \)
- \( C = (x_c, y_c) \)
- \( D = (x_d, y_d) \)
Since AD divides AB in the ratio 5:3, the coordinates of D can be found using the section formula. The section formula gives the coordinates of a point dividing a line segment in a given ratio \(m:n\) as follows:
\[
(x_d, y_d) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)
\]
Given the ratio 5:3 (m = 5, n = 3):
\[
(x_d, y_d) = \left( \frac{5 x_2 + 3 x_1}{8}, \frac{5 y_2 + 3 y_1}{8} \right)
\]
Under the condition of AC being vertical and CD being horizontal, the vertical line AC implies that \( x_c = x_a \) (no change in the x-coordinate from A to C). Similarly, the horizontal line CD implies that \(y_d = y_c\) since C and D lie on the same horizontal line.
Now, let's evaluate the answer choices based on this understanding. Consider the given coordinates options:
A. \((2, -3)\)
B. \((7, -1)\)
C. \((5, -3)\)
D. \((2, -1)\)
We don't have the specific coordinates of points A and B to utilize the method fully, yet the defining characteristics are that \(x_d = x_a\) and \(y_c = y_d\).
Since points A and C lie on a vertical line:
- The x-coordinate of point C must be equal to the x-coordinate that divides AB in the given ratio, and thus we only consider vertical adjustments.
Given the multiple choices:
- Option C: (5, -3) – This choice suggests that x_c could be related to a point on the dividing line in a plausible ratio but missing exact details about start points.
Given AC must be vertical, we focus on the horizontal displacement rule primarily.
Without exact coordinates of A and B, visualization of the problem would rely entirely on the limited given options.
Analyzing provided potential, we may infer:
- Most reasonable if input approximations best meet: \( (5, -3) \)
We choose accurate examining:
Answer is:
C. \( (5, -3)\)
In the diagram, point D divides line segment AB in the ratio of 5:3. If line segment AC is vertical and line segment CD is horizontal, what are the coordinates of point C?
A. (2,-3)
B. (7,-1)
C. (5,-3)
D. (2,-1)
1 answer
1 answer