Question

Points A, B, and C are collinear, and B is between A and C with the ratio of AB to BC being 3:4. If A is at (-2,-6) and B is at (1,-3), what are the coordinates of point C?

a) (4, -1)
b) (5, 0)
c) (3, -5)
d) (2, -2)

Answer 1

The coordinates of point C are determined by using the sectional ratio formula, considering A and C divide the line segment AC into 7 equal parts in the ratio 3:4. The calculation based on the x and y values of points A and B leads to the coordinates of point C being (4, -1).

Step-by-step explanation:

The question is asking for the coordinates of point C given that points A, B, and C are collinear, and B is between A and C with a ratio of AB to BC of 3:4. We have the coordinates for points A and B, so we can use the concept of sectional division in a ratio to find the coordinates of C. The coordinates of A are (-2,-6) and B are at (1,-3).

Since the ratio of AB to BC is 3:4, we can establish the following relations:

1. The total number of parts into which the line segment AC has been divided is 3 + 4 = 7 parts.
2. Each 'part' on the x-axis will have a length equal to (Cx - Ax)/7 where Cx and Ax are the x-coordinates of C and A respectively.
3. Each 'part' on the y-axis will have a length equal to (Cy - Ay)/7 where Cy and Ay are the y-coordinates of C and A respectively.
4. Given that B is 3 parts from A, we can express Bx as Ax + 3*((Cx - Ax)/7).
5. Similarly for By, Ay + 3*((Cy - Ay)/7).

Using the coordinates for B, we have:

• 1 = -2 + 3*((Cx + 2)/7)
• -3 = -6 + 3*((Cy + 6)/7)

Solving these equations gives Cx = 4 and Cy = -1, so the coordinates of C are (4, -1), which corresponds to option a.