Answered

The directed line segment from [tex]L[/tex] to [tex]N[/tex] has endpoints [tex]L(-6, 2)[/tex] and [tex]N(5, -3)[/tex]. What are the [tex]x[/tex]- and [tex]y[/tex]-coordinates of point [tex]M[/tex], which partitions the directed line segment into the ratio [tex]2:5[/tex]?

[tex]\[
\begin{array}{l}
x = \square \\
y = \square
\end{array}
\][/tex]



Answer :

GinnyAnswer
To determine the coordinates of point [tex]\( M \)[/tex], which partitions the directed line segment from [tex]\( L(-6,2) \)[/tex] to [tex]\( N(5,-3) \)[/tex] in the ratio [tex]\( 2:5 \)[/tex], we use the section formula for internal division. The section formula states that the coordinates of a point dividing the line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex] are given by:

[tex]\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \][/tex]
[tex]\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \][/tex]

In this case, the coordinates of [tex]\( L \)[/tex] are [tex]\((x_1, y_1) = (-6, 2)\)[/tex] and the coordinates of [tex]\( N \)[/tex] are [tex]\((x_2, y_2) = (5, -3)\)[/tex]. The ratio [tex]\( m:n \)[/tex] is [tex]\( 2:5 \)[/tex].

Now let's calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( M \)[/tex]:

[tex]\[ x = \frac{2 \cdot 5 + 5 \cdot (-6)}{2 + 5} \][/tex]
[tex]\[ x = \frac{10 - 30}{7} \][/tex]
[tex]\[ x = \frac{-20}{7} \][/tex]
[tex]\[ x = -2.857142857142857 \][/tex]

Next, let's calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( M \)[/tex]:

[tex]\[ y = \frac{2 \cdot (-3) + 5 \cdot 2}{2 + 5} \][/tex]
[tex]\[ y = \frac{-6 + 10}{7} \][/tex]
[tex]\[ y = \frac{4}{7} \][/tex]
[tex]\[ y = 0.5714285714285714 \][/tex]

Thus, the coordinates of point [tex]\( M \)[/tex] that partitions the segment [tex]\( LN \)[/tex] in the ratio [tex]\( 2:5 \)[/tex] are:

[tex]\[ x = -2.857142857142857 \checkmark \][/tex]
[tex]\[ y = 0.5714285714285714 \][/tex]

Other Questions