Question

On a coordinate plane, the endpoints of line segment AC are A(-5,-3) and C(5,7). Point B lies on line segment AC and divides it into two line segments such that the ratio of AB to BC is 3:2. What are the coordinates of point B ?

A. (-0,-2)
B. (2,4)
C. (1,3)

Answer 1

To find point B that divides the line segment AC with endpoints A(-5, -3) and C(5, 7) into a ratio of 3:2, we apply the section formula, yielding B's coordinates as (1, 3), which corresponds to option C.

Step-by-step explanation:

To find the coordinates of point B which divides the line segment AC into two parts in the ratio 3:2, we can use the section formula. The coordinates of point A are (-5, -3) and of point C are (5, 7). To apply the formula, we use the weights 3 and 2 corresponding to the two parts of the line segment that B divides.

The section formula in ratio m:n for a line divided by point P(x, y) is given by:

x = (m × x2 + n × x1)/(m + n)
y = (m × y2 + n × y1)/(m + n)

For this problem:

1. m = 3 and n = 2
2. Coordinates of A are (-5, -3) and C are (5, 7)
3. We calculate the x-coordinate of B:
   x = (3 × 5 + 2 × (-5)) / (3 + 2) = (15 - 10) / 5 = 1
4. The y-coordinate of B:
   y = (3 × 7 + 2 × (-3)) / (3 + 2) = (21 - 6) / 5 = 3

Thus, the coordinates of point B are (1, 3), which is option C.