To find the coordinates of point [tex]\( M \)[/tex] that partitions the segment with endpoints [tex]\( X(1, -2) \)[/tex] and [tex]\( Y(10, 3) \)[/tex] in a [tex]\( 5:1 \)[/tex] ratio, we can use the section formula in coordinate geometry. Here are the detailed steps to solve this problem:
1. Identify the given points and the ratio:
- Endpoint [tex]\( X \)[/tex] has coordinates [tex]\( (x_1, y_1) = (1, -2) \)[/tex].
- Endpoint [tex]\( Y \)[/tex] has coordinates [tex]\( (x_2, y_2) = (10, 3) \)[/tex].
- The ratio for partitioning is [tex]\( m:n = 5:1 \)[/tex].
2. Write down the section formula:
The section formula for the coordinates of a point [tex]\( M(x_m, y_m) \)[/tex] dividing the segment [tex]\( XY \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
x_m = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\][/tex]
[tex]\[
y_m = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
3. Substitute the values into the section formula:
- [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]
Using these values, we can calculate each coordinate of [tex]\( M \)[/tex] separately:
4. Calculate the [tex]\( x \)[/tex]-coordinate ([tex]\( x_m \)[/tex]):
[tex]\[
x_m = \frac{5 \cdot 10 + 1 \cdot 1}{5 + 1}
\][/tex]
[tex]\[
x_m = \frac{50 + 1}{6}
\][/tex]
[tex]\[
x_m = \frac{51}{6} \approx 8.5
\][/tex]
5. Calculate the [tex]\( y \)[/tex]-coordinate ([tex]\( y_m \)[/tex]):
[tex]\[
y_m = \frac{5 \cdot 3 + 1 \cdot (-2)}{5 + 1}
\][/tex]
[tex]\[
y_m = \frac{15 - 2}{6}
\][/tex]
[tex]\[
y_m = \frac{13}{6} \approx 2.1666666666666665
\][/tex]
6. Write down the coordinates of point [tex]\( M \)[/tex]:
- Thus, the coordinates of point [tex]\( M \)[/tex] are approximately [tex]\( (8.5, 2.1666666666666665) \)[/tex].
Therefore, the point [tex]\( M \)[/tex] that partitions the segment [tex]\( XY \)[/tex] in a [tex]\( 5:1 \)[/tex] ratio has coordinates [tex]\( (8.5, 2.1666666666666665) \)[/tex].
