ABCD is a kite with AB = AD and CB = CD
A is the point with coordinates (-2, 10)
B is the point with coordinates( -27/5 , 4)
C is the point with coordinates (4,−5)
Work out the coordinates of D

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.

ABCD is a kite with AB = AD and CB = CD. A is the point with coordinates (-2, 10), B is the point with coordinates (-27/5, 4), and C is the point with coordinates (4, -5). Work out the coordinates of D.

Solve for the coordinates of point D in kite ABCD, where AB = AD and CB = CD. Using properties of kites, coordinate geometry, midpoint, and slope calculations, we determine D based on known points A, B, and C. Ideal for Grades 9-10 students studying geometry and coordinate geometry.

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