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Coordinate Geometry Test Part Two

Module 04: Coordinate Geometry

Problem:

A point [tex]\( M \)[/tex] on a segment with endpoints [tex]\( X(1,-2) \)[/tex] and [tex]\( Y(10,3) \)[/tex] partitions the segment in a [tex]\( 5:1 \)[/tex] ratio. Find [tex]\( M \)[/tex]. You must show all work to receive credit. (10 points)



Answer :

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].