https://file.aiquickdraw.com/mathbot/file/1727451372146abie10tp.jpg

Let's analyze the image and solve the three math problems based on the provided formulas.

### 17. Find the distance between the points \( C(-1, 8) \) and \( D(5, 9) \). Round your answer to the nearest tenth.

The distance formula is given as:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Where:
- \( x_1 = -1 \), \( y_1 = 8 \) (coordinates of \( C \)),
- \( x_2 = 5 \), \( y_2 = 9 \) (coordinates of \( D \)).

Substituting the values:

\[
d = \sqrt{(5 - (-1))^2 + (9 - 8)^2} = \sqrt{(5 + 1)^2 + (1)^2} = \sqrt{6^2 + 1^2} = \sqrt{36 + 1} = \sqrt{37}
\]

Now, calculating \( \sqrt{37} \) to the nearest tenth:
\[
d \approx 6.1
\]

Thus, the distance between points \( C \) and \( D \) is approximately **6.1 units**.

---

### 18. Find the coordinates of the midpoint \( M \) of the segment joining \( H(2, -1) \) and \( K(3, 8) \).

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

Where:
- \( x_1 = 2 \), \( y_1 = -1 \) (coordinates of \( H \)),
- \( x_2 = 3 \), \( y_2 = 8 \) (coordinates of \( K \)).

Substituting the values:

\[
M = \left( \frac{2 + 3}{2}, \frac{-1 + 8}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) = (2.5, 3.5)
\]

Thus, the midpoint \( M \) is \( (2.5, 3.5) \).

---

### 19. The coordinates of the midpoint of segment \( AC \) are \( (2, -3) \). The coordinates of \( A \) are \( (0, 5) \). Find the coordinates of \( C \).

The midpoint formula is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Let the coordinates of \( C \) be \( (x_2, y_2) \), and we know:
- \( M = (2, -3) \),
- \( A = (0, 5) \).

Using the midpoint formula, we can set up two equations:
\[
2 = \frac{0 + x_2}{2} \quad \text{(for the x-coordinates)}
\]
\[
-3 = \frac{5 + y_2}{2} \quad \text{(for the y-coordinates)}
\]

Solve each equation:

1. \( 2 = \frac{0 + x_2}{2} \)
   \[
   x_2 = 4
   \]

2. \( -3 = \frac{5 + y_2}{2} \)
   \[
   -6 = 5 + y_2 \quad \Rightarrow \quad y_2 = -6 - 5 = -11
   \]

Thus, the coordinates of point \( C \) are \( (4, -11) \).

---

Let me know if you'd like more details or if you have any further questions. 

Here are five related questions to deepen your understanding:

1. How would the distance formula change in 3D space?
2. Can you apply the midpoint formula to find the center of a triangle?
3. What are the properties of line segments that share the same midpoint?
4. How would you calculate the slope of a line given two points?
5. How would the distance between two points change if the coordinate system was non-Cartesian?

**Tip:** Always visualize the points on a graph to better understand their spatial relationships.

Find the distance between the points C(-1, 8) and D(5, 9). Find the coordinates of the midpoint M of the segment joining H(2, -1) and K(3, 8). The coordinates of the midpoint of segment AC are (2, -3). The coordinates of A are (0, 5). Find the coordinates of C.

Learn how to calculate the distance between two points, find the midpoint of a segment, and determine missing coordinates using key concepts of coordinate geometry. This problem involves the distance formula, midpoint formula, and application of these formulas in different scenarios. Suitable for students in Grades 8-10.

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