Given that $ABCD$ is a kite with $AB = AD$ and $CB = CD$, points $A$ and $C$ are the endpoints of the kite's diagonal. In a kite, the diagonals are perpendicular bisectors of each other.

1. Find the midpoint of diagonal $AC$:
   The midpoint $M$ of $AC$ (where $M$ is also the midpoint of $BD$) is calculated by averaging the coordinates of $A$ and $C$: $M = \left(\frac{-2 + 4}{2}, \frac{10 + (-5)}{2}\right) = \left(\frac{2}{2}, \frac{5}{2}\right) = (1, 2.5)$

2. Find the slope of $AC$: The slope of $AC$ is: $\text{slope of } AC = \frac{-5 - 10}{4 - (-2)} = \frac{-15}{6} = -\frac{5}{2}$

3. Determine the slope of $BD$: Since the diagonals of a kite are perpendicular, the slope of $BD$ will be the negative reciprocal of the slope of $AC$. Thus: $\text{slope of } BD = \frac{2}{5}$

4. Set up the line equation for $BD$: The line $BD$ passes through $M = (1, 2.5)$ with slope $\frac{2}{5}$. The equation of a line is $y = mx + c$. Substituting the values: $y = \frac{2}{5}x + c$ Substitute $M$ into the equation to solve for $c$: $2.5 = \frac{2}{5}(1) + c \Rightarrow 2.5 = \frac{2}{5} + c \Rightarrow c = 2.5 - 0.4 = 2.1$ Therefore, the equation of line $BD$ is: $y = \frac{2}{5}x + 2.1$

5. Find $D$ as the reflection of $B$ across $M$: To find the coordinates of $D$, we use the midpoint formula. Since $M$ is the midpoint of $B$ and $D$: $M = \left(\frac{-\frac{27}{5} + x_D}{2}, \frac{4 + y_D}{2}\right) = (1, 2.5)$ Solving for $x_D$ and $y_D$:

   • $x$-coordinate: $\frac{-\frac{27}{5} + x_D}{2} = 1 \Rightarrow -\frac{27}{5} + x_D = 2 \Rightarrow x_D = 2 + \frac{27}{5} = \frac{37}{5} = 7.4$
   • $y$-coordinate: $\frac{4 + y_D}{2} = 2.5 \Rightarrow 4 + y_D = 5 \Rightarrow y_D = 1$

6. Conclusion: The coordinates of point $D$ are $(7.4, 1)$.

Would you like a deeper explanation on any part of the solution?

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1. What are the properties of a kite’s diagonals in geometry?
2. How do you calculate the midpoint between two points?
3. Why are the diagonals of a kite perpendicular?
4. What is the method to find the reflection point over a midpoint?
5. How does slope influence the perpendicularity of lines?

Tip: In any quadrilateral, understanding the properties of its diagonals helps in determining unknown points or angles.