Final answer:
To find the coordinates of point C that divides line segment AB in the ratio 2:1, the section formula is used, yielding the coordinates (8, 2) for point C.
Step-by-step explanation:
The student's question involves finding the coordinates of point C that divides line segment AB in the ratio 2:1. To find this point, we will apply the section formula, which is used to divide a line segment into a given ratio. Here, line segment AB has endpoints A(4,5) and B(16,-4). The coordinates of point C can be found using the formula for internal division:
Mx = (x1*m + x2*n) / (m + n),
My = (y1*m + y2*n) / (m + n),
where A(x1,y1), B(x2,y2) are the endpoints of the line segment AB, M is the point dividing the line internally in the ratio m:n, which in this case is 2:1. Substituting the given values:
Cx = (4*2 + 16*1) / (2 + 1) = (8 + 16) / 3 = 24 / 3 = 8,
Cy = (5*2 + (-4)*1) / (2 + 1) = (10 - 4) / 3 = 6 / 3 = 2,
So, the coordinates of point C are (8, 2).