To determine the coordinates of point M, which partitions the segment between points X (1, -2) and Y (10, 3) in the ratio 5:1, we will use the section formula. The section formula helps find a point dividing a line segment internally in a given ratio.
Given:
- Point X has coordinates (1, -2).
- Point Y has coordinates (10, 3).
- The ratio in which point M divides the segment XY is 5:1. This means [tex]\( m = 5 \)[/tex] and [tex]\( n = 1 \)[/tex].
The section formula for the coordinates of point M [tex]\((Mx, My)\)[/tex] which divides the line segment internally in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ Mx = \frac{m \cdot Xx + n \cdot Yx}{m + n} \][/tex]
[tex]\[ My = \frac{m \cdot Xy + n \cdot Yy}{m + n} \][/tex]
Substituting the given values into the formulas:
1. Calculate the x-coordinate of point M (Mx):
[tex]\[ Mx = \frac{5 \cdot 1 + 1 \cdot 10}{5 + 1} \][/tex]
[tex]\[ Mx = \frac{5 + 10}{6} \][/tex]
[tex]\[ Mx = \frac{15}{6} \][/tex]
[tex]\[ Mx = 2.5 \][/tex]
2. Calculate the y-coordinate of point M (My):
[tex]\[ My = \frac{5 \cdot (-2) + 1 \cdot 3}{5 + 1} \][/tex]
[tex]\[ My = \frac{-10 + 3}{6} \][/tex]
[tex]\[ My = \frac{-7}{6} \][/tex]
[tex]\[ My = -1.1666666666666667 \][/tex]
Thus, the coordinates of point M are:
[tex]\[ M = (2.5, -1.1666666666666667) \][/tex]
Therefore, point M, which partitions the segment joining points X and Y in the ratio 5:1, has coordinates (2.5, -1.1666666666666667).
