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Let's go through the questions one by one. I'll solve each one for you with explanations.

### 18. \( L(-3, 16), M(17, 4) \)
The formula for the midpoint between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Here, the points are \( (-3, 16) \) and \( (17, 4) \):
\[
\left( \frac{-3 + 17}{2}, \frac{16 + 4}{2} \right) = \left( \frac{14}{2}, \frac{20}{2} \right) = (7, 10)
\]
The midpoint is \( (7, 10) \).

### 19. \( C(32, -1), D(0, -12) \)
Using the same formula for midpoint:
\[
\left( \frac{32 + 0}{2}, \frac{-1 + (-12)}{2} \right) = \left( \frac{32}{2}, \frac{-13}{2} \right) = (16, -6.5)
\]
The midpoint is \( (16, -6.5) \).

### 20. \( X(-11, -6), M(15, 4) \)
Here, \( M(15, 4) \) is the midpoint of \( XY \), and we are given one endpoint \( X(-11, -6) \).

We use the midpoint formula in reverse to find \( Y(x_2, y_2) \). Since \( M \) is the midpoint:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

We know that:
\[
15 = \frac{-11 + x_2}{2} \quad \text{and} \quad 4 = \frac{-6 + y_2}{2}
\]

Solve for \( x_2 \):
\[
15 \times 2 = -11 + x_2 \quad \Rightarrow \quad 30 = -11 + x_2 \quad \Rightarrow \quad x_2 = 41
\]

Solve for \( y_2 \):
\[
4 \times 2 = -6 + y_2 \quad \Rightarrow \quad 8 = -6 + y_2 \quad \Rightarrow \quad y_2 = 14
\]

Thus, \( Y(41, 14) \).

### 21. \( M(-4, 8), Y(19, 0) \)
Given the midpoint \( M(-4, 8) \) and one endpoint \( Y(19, 0) \), we need to find the other endpoint \( X(x_1, y_1) \).

Using the formula for the midpoint:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Here:
\[
-4 = \frac{x_1 + 19}{2} \quad \text{and} \quad 8 = \frac{y_1 + 0}{2}
\]

Solve for \( x_1 \):
\[
-4 \times 2 = x_1 + 19 \quad \Rightarrow \quad -8 = x_1 + 19 \quad \Rightarrow \quad x_1 = -27
\]

Solve for \( y_1 \):
\[
8 \times 2 = y_1 + 0 \quad \Rightarrow \quad y_1 = 16
\]

Thus, \( X(-27, 16) \).

### 22. Hiking Problem
Given Carol's location \( (7, 13) \) and Marita's location \( (3, 5) \):

#### a. Find the coordinates of the point midway between their locations.
We use the midpoint formula:
\[
\left( \frac{7 + 3}{2}, \frac{13 + 5}{2} \right) = \left( \frac{10}{2}, \frac{18}{2} \right) = (5, 9)
\]
The midpoint is \( (5, 9) \).

#### b. Find the coordinates of the point \( \frac{1}{3} \) the distance from Carol to Marita.
We can use a weighted average. To find a point \( \frac{1}{3} \) from Carol toward Marita:
\[
x = 7 + \frac{1}{3} (3 - 7) = 7 + \frac{1}{3} (-4) = 7 - \frac{4}{3} = \frac{21}{3} - \frac{4}{3} = \frac{17}{3}
\]
\[
y = 13 + \frac{1}{3} (5 - 13) = 13 + \frac{1}{3} (-8) = 13 - \frac{8}{3} = \frac{39}{3} - \frac{8}{3} = \frac{31}{3}
\]
The point is \( \left( \frac{17}{3}, \frac{31}{3} \right) \).

Would you like detailed explanations or further clarifications for any of these? 

### 5 Relative Questions:
1. How do you find the coordinates of a point at any fractional distance between two given points?
2. What is the formula to find the missing endpoint when the midpoint and one endpoint are given?
3. How can you apply the midpoint formula to problems involving weighted distances, like \( \frac{1}{3} \) from a point?
4. How would the formula change if we were working in 3D space?
5. Can you calculate the distance between Carol and Marita after finding their midpoint?

### Tip:
For any problem involving fractions of distance between two points, you can generalize the method using the formula:  
\[
P = (x_1 + r(x_2 - x_1), y_1 + r(y_2 - y_1))
\]
where \( r \) is the fraction of the distance.

Find the coordinates of the midpoint of a segment with the given endpoints: 18. L(-3, 16), M(17, 4); 19. C(32, -1), D(0, -12); Find the coordinates of the missing endpoint if M is the midpoint of XY: 20. X(-11, -6), M(15, 4); 21. M(-4, 8), Y(19, 0); 22. Carol and Marita are hiking in a state park and decide to take separate trails. The map of the park is set up on a coordinate grid. Carol's location is at the point (7, 13) and Marita is at (3, 5). a) Find the coordinates of the point midway between their locations. b) Find the coordinates of the point 1/3 the distance from Carol to Marita.

Solve problems related to finding midpoints, missing endpoints, and fractional distances on a coordinate plane. Includes step-by-step explanations and formulas, ideal for students in Grades 8-10.

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