Question

Consider points A(8,1) and B(12,3). Find point C on the x-axis so AC + BC is a minimum.

(A) (9,0)
(B) (10,0)
(C) (11,0)
(D) (12,0)

Answer 1

To find point C on the x-axis such that the sum of AC and BC is a minimum, we can minimize the distances AC and BC. By finding the x-coordinate of point C that minimizes AC + BC, we determine that the point C on the x-axis is (10, 0).

Step-by-step explanation:

To find point C on the x-axis such that the sum of AC and BC is a minimum, we need to minimize the distances AC and BC. Since point C is on the x-axis, its y-coordinate will be 0. Let's denote the x-coordinate of point C as x.

The distance AC can be found using the distance formula: AC = sqrt((x - 8)^2 + (0 - 1)^2)

The distance BC can also be found using the distance formula: BC = sqrt((x - 12)^2 + (0 - 3)^2)

To find the value of x that minimizes the sum of AC and BC, we can find the first derivative of the sum with respect to x, set it equal to 0, and solve for x. Alternately, we can graph the sum of AC and BC as a function of x and find the minimum point on the graph. After performing the calculations, we find that the point C on the x-axis that minimizes AC + BC is (10, 0). Therefore, the correct answer is (B) (10, 0).