To find the midpoint of a line segment with given endpoints, we use the midpoint formula. The midpoint formula states that the midpoint [tex]\((M)\)[/tex] of a segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Let’s apply this formula to our specific endpoints [tex]\((a-2, b)\)[/tex] and [tex]\((a+4, b)\)[/tex].
1. Identify the coordinates of the endpoints:
- Endpoint 1: [tex]\((x_1, y_1) = (a-2, b)\)[/tex]
- Endpoint 2: [tex]\((x_2, y_2) = (a+4, b)\)[/tex]
2. Apply the midpoint formula:
[tex]\[
M = \left( \frac{(a-2) + (a+4)}{2}, \frac{b + b}{2} \right)
\][/tex]
3. Simplify the coordinates:
- For the x-coordinate:
[tex]\[
\frac{(a-2) + (a+4)}{2} = \frac{a - 2 + a + 4}{2} = \frac{2a + 2}{2} = a + 1
\][/tex]
- For the y-coordinate:
[tex]\[
\frac{b + b}{2} = \frac{2b}{2} = b
\][/tex]
4. Therefore, the coordinates of the midpoint are:
[tex]\[
M = (a + 1, b)
\][/tex]
So, the midpoint of the segment with endpoints [tex]\((a-2, b)\)[/tex] and [tex]\((a+4, b)\)[/tex] is [tex]\((a+1, b)\)[/tex].
