Mathematics
Mathematics
Problem
The points and are the endpoints of a line segment. A point divides internally and is known to lie on the -axis. a. (i) Determine, in the simplest integer form, the ratio in which divides . b. (ii) Find the coordinates of the point . c. (iii) Verify your answer by showing that the ratio of the lengths equals the ratio obtained in part (i).Intuitive explanation
The key is recognizing that any point on the -axis has coordinates for some value . Since divides internally, it must satisfy the section formula—meaning it splits the segment in some ratio . By setting the -coordinate of the section formula equal to zero, you can find that ratio, then use it to find the -coordinate. This turns a geometry problem into an algebra problem about weighted averages.
Intuitive explanation
The key is recognizing that any point on the -axis has coordinates for some value . Since divides internally, it must satisfy the section formula—meaning it splits the segment in some ratio . By setting the -coordinate of the section formula equal to zero, you can find that ratio, then use it to find the -coordinate. This turns a geometry problem into an algebra problem about weighted averages.