To solve the problem of finding the point [tex]\( M \)[/tex] that partitions the segment between points [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] in the ratio 5:1, we can utilize the section formula. The section formula is a method in coordinate geometry used to find the coordinates of a point that divides a line segment joining two given points in a given ratio.
Given points:
- [tex]\( X(1, -2) \)[/tex]
- [tex]\( Y(10, 3) \)[/tex]
The ratio given is 5:1, i.e., [tex]\( m:n = 5:1 \)[/tex].
The section formula states that if a point [tex]\( M \)[/tex] divides the line segment joining two points [tex]\( X(x_1, y_1) \)[/tex] and [tex]\( Y(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( M(x, y) \)[/tex] are given by:
[tex]\[ x = \frac{m x_2 + n x_1}{m + n} \][/tex]
[tex]\[ y = \frac{m y_2 + n y_1}{m + n} \][/tex]
Here,
- [tex]\( x_1 = 1 \)[/tex]
- [tex]\( y_1 = -2 \)[/tex]
- [tex]\( x_2 = 10 \)[/tex]
- [tex]\( y_2 = 3 \)[/tex]
- [tex]\( m = 5 \)[/tex]
- [tex]\( n = 1 \)[/tex]
Let's calculate the coordinates [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for point [tex]\( M \)[/tex]:
1. Calculate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{5 \cdot 10 + 1 \cdot 1}{5 + 1} = \frac{50 + 1}{6} = \frac{51}{6} = 8.5
\][/tex]
2. Calculate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{5 \cdot 3 + 1 \cdot -2}{5 + 1} = \frac{15 - 2}{6} = \frac{13}{6} \approx 2.1667
\][/tex]
Therefore, the coordinates of point [tex]\( M \)[/tex] that partitions the segment [tex]\( XY \)[/tex] in the ratio 5:1 are:
[tex]\[ M(8.5, 2.1667) \][/tex]
In summary, the point [tex]\( M \)[/tex] that divides the segment with endpoints [tex]\( X(1, -2) \)[/tex] and [tex]\( Y(10, 3) \)[/tex] in a 5:1 ratio is located at coordinates [tex]\( (8.5, 2.1667) \)[/tex].
