Show that the points A(2,3,5),B(-4,7,-7),C(-2,1,-10) and D(4,-3,2) are the vertices of a rectangle.

Explanation

To show that the points A(2, 3, 5), B(-4, 7, -7), C(-2, 1, -10), and D(4, -3, 2) are the vertices of a rectangle, we need to prove that the opposite sides are equal and the diagonals are equal.

### Step 1: Calculate the distances AB and CD

Using the distance formula between two points in three-dimensional space, the distance \(d\) between points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

**Calculating AB:**
- \(A(2, 3, 5)\) and \(B(-4, 7, -7)\)
\[ AB = \sqrt{((-4) - 2)^2 + (7 - 3)^2 + ((-7) - 5)^2} \]
\[ = \sqrt{(-6)^2 + (4)^2 + (-12)^2} \]
\[ = \sqrt{36 + 16 + 144} \]
\[ = \sqrt{196} = 14 \]

**Calculating CD:**
- \(C(-2, 1, -10)\) and \(D(4, -3, 2)\)
\[ CD = \sqrt{(4 - (-2))^2 + ((-3) - 1)^2 + (2 - (-10))^2} \]
\[ = \sqrt{(6)^2 + (-4)^2 + (12)^2} \]
\[ = \sqrt{36 + 16 + 144} \]
\[ = \sqrt{196} = 14 \]

### Step 2: Calculate the distances AD and BC

**Calculating AD:**
- \(A(2, 3, 5)\) and \(D(4, -3, 2)\)
\[ AD = \sqrt{(4 - 2)^2 + ((-3) - 3)^2 + (2 - 5)^2} \]
\[ = \sqrt{(2)^2 + (-6)^2 + (-3)^2} \]
\[ = \sqrt{4 + 36 + 9} \]
\[ = \sqrt{49} = 7 \]

**Calculating BC:**
- \(B(-4, 7, -7)\) and \(C(-2, 1, -10)\)
\[ BC = \sqrt{((-2) - (-4))^2 + (1 - 7)^2 + ((-10) - (-7))^2} \]
\[ = \sqrt{(2)^2 + (-6)^2 + (-3)^2} \]
\[ = \sqrt{4 + 36 + 9} \]
\[ = \sqrt{49} = 7 \]

### Step 3: Calculate the distances AC and BD (diagonals)

**Calculating AC:**
- \(A(2, 3, 5)\) and \(C(-2, 1, -10)\)
\[ AC = \sqrt{((-2) - 2)^2 + (1 - 3)^2 + ((-10) - 5)^2} \]
\[ = \sqrt{(-4)^2 + (-2)^2 + (-15)^2} \]
\[ = \sqrt{16 + 4 + 225} \]
\[ = \sqrt{245} = 7\sqrt{5} \]

**Calculating BD:**
- \(B(-4, 7, -7)\) and \(D(4, -3, 2)\)
\[ BD = \sqrt{(4 - (-4))^2 + ((-3) - 7)^2 + (2 - (-7))^2} \]
\[ = \sqrt{(8)^2 + (-10)^2 + (9)^2} \]
\[ = \sqrt{64 + 100 + 81} \]
\[ = \sqrt{245} = 7\sqrt{5} \]

### Conclusion

We have shown that:
- \(AB = CD = 14\)
- \(AD = BC = 7\)
- \(AC = BD = 7\sqrt{5}\)

Since opposite sides are equal and the diagonals are equal, the points A, B, C, and D form a rectangle.