To determine the coordinates of point [tex]\( M \)[/tex], we need to go through a step-by-step process of finding the intermediate point [tex]\( L \)[/tex] and then [tex]\( M \)[/tex].
### Step 1: Finding Point [tex]\( L \)[/tex]
Point [tex]\( L \)[/tex] partitions the directed line segment [tex]\( K(-6,-2) \)[/tex] to [tex]\( N(8,3) \)[/tex] in the ratio [tex]\( 1:2 \)[/tex].
- The formula for finding the coordinates [tex]\((x, y)\)[/tex] of a point that divides a segment in the ratio [tex]\( m:n \)[/tex] is:
[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]
- Plugging the values for point [tex]\( L \)[/tex]:
- [tex]\( K = (x_1, y_1) = (-6, -2) \)[/tex]
- [tex]\( N = (x_2, y_2) = (8, 3) \)[/tex]
- [tex]\( m = 1 \)[/tex], [tex]\( n = 2 \)[/tex]
[tex]\[
l_x = \frac{1 \times 8 + 2 \times (-6)}{1+2} = \frac{8 - 12}{3} = \frac{-4}{3} = -1.3333
\][/tex]
[tex]\[
l_y = \frac{1 \times 3 + 2 \times (-2)}{1+2} = \frac{3 - 4}{3} = \frac{-1}{3} = -0.3333
\][/tex]
So, the coordinates of point [tex]\( L \)[/tex] are approximately [tex]\((-1.3333, -0.3333)\)[/tex].
### Step 2: Finding Point [tex]\( M \)[/tex]
Point [tex]\( M \)[/tex] partitions the directed line segment from [tex]\( L(-1.3333, -0.3333) \)[/tex] to [tex]\( N(8, 3) \)[/tex] in the ratio [tex]\( 3:1 \)[/tex].
- Using the same formula as before:
[tex]\[
m_x = \frac{3 \cdot 8 + 1 \cdot (-1.3333)}{3+1} = \frac{24 - 1.3333}{4} = \frac{22.6667}{4} = 5.6667
\][/tex]
[tex]\[
m_y = \frac{3 \cdot 3 + 1 \cdot (-0.3333)}{3+1} = \frac{9 - 0.3333}{4} = \frac{8.6667}{4} = 2.1667
\][/tex]
### Step 3: Rounding to the Nearest Tenth
Finally, we round the coordinates of point [tex]\( M \)[/tex] to the nearest tenth:
- [tex]\( m_x \approx 5.7 \)[/tex]
- [tex]\( m_y \approx 2.2 \)[/tex]
Thus, the coordinates of point [tex]\( M \)[/tex] are [tex]\( (5.7, 2.2) \)[/tex].
So, the correct answer is:
[tex]\[
(5.7, 2.2)
\][/tex]
