Midpoints and Segment Congruence Study Guide with practice problems and coordinate geometry examples.

Problem Analysis:

The task involves finding the coordinates of the midpoint of a segment given the endpoints. The formula for the midpoint of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

We are provided with two segments, \(IH\) and \(HJ\), and their respective endpoints. We need to calculate the midpoints of these segments using the given formula.

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Step-by-Step Solution:

#### Segment 1: \(IH\)
- Endpoints: \(I(3, 4)\) and \(H(-1, -2)\)
- Using the midpoint formula:
\[
\text{Midpoint of } IH = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substituting the coordinates of \(I\) and \(H\):
\[
\text{Midpoint of } IH = \left( \frac{3 + (-1)}{2}, \frac{4 + (-2)}{2} \right)
\]
Simplify each component:
\[
\text{Midpoint of } IH = \left( \frac{3 - 1}{2}, \frac{4 - 2}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1)
\]

#### Segment 2: \(HJ\)
- Endpoints: \(H(-1, -2)\) and \(J(5, 0)\)
- Using the midpoint formula:
\[
\text{Midpoint of } HJ = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substituting the coordinates of \(H\) and \(J\):
\[
\text{Midpoint of } HJ = \left( \frac{-1 + 5}{2}, \frac{-2 + 0}{2} \right)
\]
Simplify each component:
\[
\text{Midpoint of } HJ = \left( \frac{4}{2}, \frac{-2}{2} \right) = (2, -1)
\]

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Final Answers:

- The midpoint of segment \(IH\) is \((1, 1)\).
- The midpoint of segment \(HJ\) is \((2, -1)\).

Thus, the solutions are:
\[
\boxed{(1, 1) \text{ and } (2, -1)}
\]