To find the coordinates of point M that partitions the line segment from point L to point N in the given ratio of 2:5, we use the section formula. The section formula for dividing a line segment in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
M = \left( \frac{n x_1 + m x_2}{m + n}, \frac{n y_1 + m y_2}{m + n} \right)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] are the coordinates of [tex]\( L \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of [tex]\( N \)[/tex].
Given:
- Coordinates of [tex]\( L \)[/tex] are [tex]\( (-6, 2) \)[/tex]
- Coordinates of [tex]\( N \)[/tex] are [tex]\( (5, -3) \)[/tex]
- Ratio [tex]\( m:n \)[/tex] is [tex]\( 2:5 \)[/tex]
Let [tex]\( m = 2 \)[/tex] and [tex]\( n = 5 \)[/tex].
We now substitute these values into the section formula.
For the x-coordinate of point M:
[tex]\[
x_M = \frac{n \cdot x_L + m \cdot x_N}{m + n} = \frac{5 \cdot (-6) + 2 \cdot 5}{2 + 5}
\][/tex]
Calculation:
[tex]\[
x_M = \frac{(5 \cdot -6) + (2 \cdot 5)}{7} = \frac{-30 + 10}{7} = \frac{-20}{7} \approx -2.8571
\][/tex]
For the y-coordinate of point M:
[tex]\[
y_M = \frac{n \cdot y_L + m \cdot y_N}{m + n} = \frac{5 \cdot 2 + 2 \cdot (-3)}{2 + 5}
\][/tex]
Calculation:
[tex]\[
y_M = \frac{(5 \cdot 2) + (2 \cdot -3)}{7} = \frac{10 - 6}{7} = \frac{4}{7} \approx 0.5714
\][/tex]
Thus, the coordinates of point M are:
[tex]\[
(x, y) = (-2.8571, 0.5714)
\][/tex]
Therefore, the x-coordinate of point [tex]\( M \)[/tex] is [tex]\(-2.8571\)[/tex] and the y-coordinate of point [tex]\( M \)[/tex] is [tex]\( 0.5714 \)[/tex].
