To find the missing coordinates of points K, L, and M that lie on the same line as points A, B, C, D, and E, we first need to determine the equation of the line that passes through at least two of those points.
Let's consider points A (2, 3) and B (4, 6):
- Calculate the slope \( m \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 3}{4 - 2} = \frac{3}{2} \]
- Use point-slope form to write the equation of the line. Using point A (2, 3):
\[ y - y_1 = m(x - x_1) \implies y - 3 = \frac{3}{2}(x - 2) \]
- Simplifying this equation gives:
\[ y - 3 = \frac{3}{2}x - 3 \implies y = \frac{3}{2}x \]
Now we can find the missing coordinates of points K, L, and M using this equation.
- For point K (10, ...):
\[ y = \frac{3}{2}(10) = 15 \] So, K (10, 15).
- For point L (..., -18):
Setting \( y = -18 \) in the line equation:
\[ -18 = \frac{3}{2}x \] Solving for \( x \):
\[ x = -18 \cdot \frac{2}{3} = -12 \] So, L (-12, -18).
- For point M (30, ...):
\[ y = \frac{3}{2}(30) = 45 \] So, M (30, 45).
Finally, the coordinates are:
- K (10, 15)
- L (-12, -18)
- M (30, 45)