Final answer:
The two possible locations for point C can be found using the section formula, dividing line segment AB in the ratio of 1:4. This division can be translated to coordinates by applying the formula for each scenario, where C is either closer to A or to B, giving us the two possible sets of coordinates.
Step-by-step explanation:
There are two possible locations for point C on line segment AB which divide it in the ratio of 1:4—either C is closer to A or closer to B.
To find the coordinates of point C, we can use the section formula which is applied in the ratio of division. Since point C divides line segment AB in the ratio of 1:4, we treat one of the divisions as 1x and the other as 4x, where 'x' is the common unit. Let's proceed with a step by step explanation.
First, we find the total length of AB in terms of 'x', which is 1x + 4x = 5x. Then, we can determine the distances that point C is placed from point A in each scenario: If C is closer to A, it will be at a distance of 1x from A; if C is closer to B, it will be at a distance of 4x from A. In both cases, we use the formula for finding the x and y coordinates of C:
- The x-coordinate of C = (1 * xB + 4 * xA) / 5 = (1 * 14 + 4 * (-1)) / 5
- The y-coordinate of C = (1 * yB + 4 * yA) / 5 = (1 * 2 + 4 * (-3)) / 5
By solving these equations, we get the two possible sets of coordinates for point C.