Let's solve the problem step-by-step:
1. Identify the coordinates and the ratio:
- The coordinates of point X are [tex]\( X = (1, -2) \)[/tex].
- The coordinates of point Y are [tex]\( Y = (10, 3) \)[/tex].
- The ratio in which point M divides the segment is 5:1, meaning [tex]\( m = 5 \)[/tex] and [tex]\( n = 1 \)[/tex].
2. Apply the section formula:
The section formula in a 2-dimensional plane for a point dividing a line segment in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
M = \left( \frac{m \cdot X_2 + n \cdot X_1}{m + n}, \frac{m \cdot Y_2 + n \cdot Y_1}{m + n} \right)
\][/tex]
Where:
- [tex]\( (X_1, Y_1) \)[/tex] are the coordinates of point X.
- [tex]\( (X_2, Y_2) \)[/tex] are the coordinates of point Y.
- [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the ratio parts.
3. Substitute the coordinates and ratio into the formula:
- From the given data:
[tex]\[ m = 5 \][/tex]
[tex]\[ n = 1 \][/tex]
[tex]\[ X_1 = 1 \][/tex]
[tex]\[ Y_1 = -2 \][/tex]
[tex]\[ X_2 = 10 \][/tex]
[tex]\[ Y_2 = 3 \][/tex]
- Substituting these values into the formula, we get:
[tex]\[
M_x = \frac{(5 \cdot 10) + (1 \cdot 1)}{5 + 1} = \frac{50 + 1}{6} = \frac{51}{6} = 8.5
\][/tex]
[tex]\[
M_y = \frac{(5 \cdot 3) + (1 \cdot -2)}{5 + 1} = \frac{15 + (-2)}{6} = \frac{13}{6} \approx 2.1666666666666665
\][/tex]
4. Write the coordinates of point M:
Hence, the coordinates of point M are:
[tex]\[
M = (8.5, 2.1666666666666665)
\][/tex]
So, point M which divides the segment connecting points X (1, -2) and Y (10, 3) in the ratio 5:1 is [tex]\( (8.5, 2.1666666666666665) \)[/tex].
