91Ó°ÊÓ

Coordinates are a set of values that show an exact position. In the context of geometry, coordinates are typically represented as pairs like \( (x, y) \) to mark positions on a two-dimensional plane.
For example, the endpoints of the line segment in this exercise are given as \( (\frac{\pi}{2}, 1) \) and \( (\pi, 1) \).
The first value in each pair (like \( \frac{\pi}{2} \) and \( \pi \)) refers to the position along the X-axis, while the second value (which is \( 1 \) in both cases here) refers to the position along the Y-axis. These coordinated points are fundamental as they allow us to define the line segment and use formulas like the midpoint formula effectively.

A line segment is a part of a line that is bounded by two distinct endpoints. It includes all the points on the line between these endpoints.
In this exercise, the line segment lies between the points \( \left( \frac{\pi}{2}, 1 \right) \) and \( (\pi, 1) \).
The line segment provides the necessary structure for various geometric calculations, including finding the midpoint. To find the midpoint, you need to have clear information about both endpoints of the segment.

Simplification is the process of making an equation or expression easier to understand and use.
In the context of this exercise, once we substitute the coordinates of the endpoints into the midpoint formula, we need to simplify the resulting expressions.
For example, we start with:
\( \left( \frac{\frac{\pi}{2} + \pi}{2}, \frac{1 + 1}{2} \right) \)
To simplify this, for the \( x \) coordinate, we combine and divide suitably:
\( \frac{\frac{\pi}{2} + \pi}{2} = \frac{\frac{3\pi}{2}}{2} = \frac{3\pi}{4} \)
The \( y \) coordinate simplifies as:
\( \frac{1 + 1}{2} = \frac{2}{2} = 1\)
Thus, the midpoint is situated at \( \left( \frac{3\pi}{4}, 1 \right) \). Simplifying these steps allows us to clearly interpret the final midpoint values.