A line segment has an endpoint at (4, 6) and the midpoint of the line segment is (1, 5). What are the coordinates of the point at the other end of the line segment?

To find the coordinates of the other endpoint of the line segment, we can use the formula for the midpoint of a line segment. The midpoint, M, of a line segment with endpoints A(x₁, y₁) and B(x₂, y₂) is given by:

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

In this case, we know one endpoint A is (4, 6) and the midpoint M is (1, 5). Let’s denote the coordinates of the other endpoint B as (x, y).

Substituting the known values into the midpoint formula:

• For the x-coordinates: 1 = \frac{4 + x}{2}
• For the y-coordinates: 5 = \frac{6 + y}{2}

Now we can solve these equations one by one.

Starting with the x-coordinate:

1 = \frac{4 + x}{2}

Multiplying both sides by 2:

2 = 4 + x

Subtracting 4 from both sides gives:

x = 2 - 4 = -2

Next, we solve for the y-coordinate:

5 = \frac{6 + y}{2}

Again, multiplying both sides by 2 gives:

10 = 6 + y

Subtracting 6 from both sides results in:

y = 10 - 6 = 4

Therefore, the coordinates of the other endpoint B are (-2, 4).

In conclusion, with one endpoint at (4, 6) and a midpoint at (1, 5), the coordinates of the other end of the line segment are (-2, 4).