Question

Kerri is writing code for a video game where a ball travels back and forth across a line segment. The ball starts at the coordinate point (2,4) and travels halfway across the segment to (5,5). What is the coordinate point of the other endpoint of the segment?

(2, 3.5)
(1,8)
(8,6)
(0,10)

Answer 1

To find the other endpoint of the segment, we can illustrate the situation as follows:

Let the starting point of the ball be \( A(2, 4) \) and the point it travels to be \( B(5, 5) \). Since the ball travels halfway to the other endpoint, we can denote the other endpoint as \( C(x, y) \).

The midpoint \( M \) of a line segment connecting points \( A(x_1, y_1) \) and \( C(x_2, y_2) \) is calculated by the formula:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
In our case, the midpoint \( M \) is also the point \( B(5, 5) \):

\[
B = M = \left( \frac{2 + x}{2}, \frac{4 + y}{2} \right)
\]

We can now set the x-coordinates equal and y-coordinates equal to find \( x \) and \( y \).

### For the x-coordinates:
\[
\frac{2 + x}{2} = 5
\]
Multiplying both sides by 2 to eliminate the fraction:
\[
2 + x = 10
\]
Subtracting 2 from both sides:
\[
x = 8
\]

### For the y-coordinates:
\[
\frac{4 + y}{2} = 5
\]
Again, multiplying both sides by 2:
\[
4 + y = 10
\]
Subtracting 4 from both sides:
\[
y = 6
\]

Therefore, the coordinates of the other endpoint \( C \) are \( (8, 6) \).

The correct answer is:
\[
\boxed{(8, 6)}
\]