To solve this problem, let's identify the given information and the formula we need.

Given:

• $M = (-4, 8)$ is the midpoint of line segment $DE$.
• Point $D$ has coordinates $(6, 1)$.
• We need to find the coordinates of point $E$.

Midpoint Formula:

The midpoint $M$ of a line segment connecting two points $D(x_1, y_1)$ and $E(x_2, y_2)$ is given by: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Apply the Midpoint Formula:

Since $M = (-4, 8)$, $D = (6, 1)$, and $E = (x_2, y_2)$, we can set up equations to solve for $x_2$ and $y_2$.

1. For the x-coordinate: $-4 = \frac{6 + x_2}{2}$

2. For the y-coordinate: $8 = \frac{1 + y_2}{2}$

Solve for $x_2$ and $y_2$:

1. Solve for $x_2$: $-4 = \frac{6 + x_2}{2}$

Multiply both sides by 2: $-8 = 6 + x_2$

Subtract 6 from both sides: $x_2 = -8 - 6 = -14$

2. Solve for $y_2$: $8 = \frac{1 + y_2}{2}$

Multiply both sides by 2: $16 = 1 + y_2$

Subtract 1 from both sides: $y_2 = 16 - 1 = 15$

Coordinates of $E$:

The coordinates of point $E$ are $(-14, 15)$.

Would you like further details or have any questions?

Related Questions:

1. How would the solution change if $D$ and $M$ coordinates were different?
2. What if $M$ were not the midpoint but rather some other ratio between $D$ and $E$?
3. How can the distance formula be applied to verify the coordinates of $E$?
4. Can we find $E$ if only the distance between $D$ and $E$ was given instead of the midpoint?
5. How does this method change in three-dimensional space?

Tip:

When solving for coordinates, always check your work by substituting back into the original equations to ensure they are satisfied.