Midpoint and Distance Formulas worksheet with definitions, formulas, and exercises.

Problem Analysis:

The task involves solving problems related to the midpoint formula and its applications in geometry. The midpoint formula is used to find the coordinates of the midpoint of a line segment given the coordinates of its endpoints. The formula is:

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

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the endpoints of the segment.

Step-by-Step Solution:

#### Part 1: Understanding the Midpoint Formula
The midpoint formula divides a line segment into two equal parts. Given two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the coordinates of the midpoint \( M \) are calculated as:

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

#### Part 2: Solving Specific Problems

##### Problem a: Find the midpoint of \( \overline{RS} \) where the coordinates are \( R(9, -3) \) and \( S(-5, 5) \).
- Given: \( R(9, -3) \) and \( S(-5, 5) \)
- Formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
- Substitute the coordinates:
\[
M = \left( \frac{9 + (-5)}{2}, \frac{-3 + 5}{2} \right)
\]
- Simplify:
\[
M = \left( \frac{4}{2}, \frac{2}{2} \right) = (2, 1)
\]

Answer for Problem a: \( M = (2, 1) \)

##### Problem b: Find the midpoint \( M \) given the coordinates of \( C(-3, 8) \) and \( D(7, -6) \).
- Given: \( C(-3, 8) \) and \( D(7, -6) \)
- Formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
- Substitute the coordinates:
\[
M = \left( \frac{-3 + 7}{2}, \frac{8 + (-6)}{2} \right)
\]
- Simplify:
\[
M = \left( \frac{4}{2}, \frac{2}{2} \right) = (2, 1)
\]

Answer for Problem b: \( M = (2, 1) \)

##### Problem c: Find the midpoint given the coordinates of endpoints \( (-1, -7) \) and \( (3, 7) \).
- Given: Endpoints \( (-1, -7) \) and \( (3, 7) \)
- Formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
- Substitute the coordinates:
\[
M = \left( \frac{-1 + 3}{2}, \frac{-7 + 7}{2} \right)
\]
- Simplify:
\[
M = \left( \frac{2}{2}, \frac{0}{2} \right) = (1, 0)
\]

Answer for Problem c: \( M = (1, 0) \)

##### Problem d: Find the endpoint \( S \) given explicit point \( R(-3, 5) \) and midpoint \( M(0, 1) \).
- Given: Endpoint \( R(-3, 5) \), midpoint \( M(0, 1) \), and need to find endpoint \( S(x_2, y_2) \).
- Midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
- Substitute known values:
\[
(0, 1) = \left( \frac{-3 + x_2}{2}, \frac{5 + y_2}{2} \right)
\]
- Solve for \( x_2 \):
\[
0 = \frac{-3 + x_2}{2} \implies 0 \cdot 2 = -3 + x_2 \implies x_2 = 3
\]
- Solve for \( y_2 \):
\[
1 = \frac{5 + y_2}{2} \implies 1 \cdot 2 = 5 + y_2 \implies 2 = 5 + y_2 \implies y_2 = -3
\]

Answer for Problem d: \( S = (3, -3) \)

Final Answers:

\[
\boxed{(2, 1), (2, 1), (1, 0), (3, -3)}
\]