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.