To complete the rectangle based on these three points, let's examine the properties of rectangles:

1. Rectangles have four right angles.
2. Opposite sides are both parallel and equal in length.

Given points:

• $A = (-5, -5)$
• $B = (1, 4)$
• $C = (4, 2)$

Step-by-Step Solution

To determine the fourth vertex, let’s identify which sides of the rectangle we have:

1. Calculate the distances between the points:
   • $AB = \sqrt{(1 - (-5))^2 + (4 - (-5))^2} = \sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117}$
   • $BC = \sqrt{(4 - 1)^2 + (2 - 4)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}$
   • $AC = \sqrt{(4 - (-5))^2 + (2 - (-5))^2} = \sqrt{9^2 + 7^2} = \sqrt{81 + 49} = \sqrt{130}$

This tells us that $AB$ and $AC$ are not equal, so $AB$ and $BC$ represent different side lengths of the rectangle.

2. Find the fourth vertex, $D$: Since rectangles have parallel, equal-length sides, $D$ should be the point where a line parallel to $AB$ passes through $C$, and a line parallel to $BC$ passes through $A$.

3. Calculate $D$: The coordinates of $D$ will be at: $D = (-5 + (4 - 1), -5 + (2 - 4)) = (-5 + 3, -5 - 2) = (-2, -7)$

Solution:

The fourth vertex needed to complete the rectangle is: $D = (-2, -7)$

The rectangle vertices are $(-5, -5)$, $(1, 4)$, $(4, 2)$, and $(-2, -7)$.

Would you like further details or have any questions?

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Related Questions:

1. How can I determine if four points form a rectangle on a graph?
2. How would this approach change if we were finding vertices of a square instead?
3. What are ways to calculate the area of a rectangle given its vertices?
4. How can one confirm if two line segments are parallel using points?
5. How would I use the midpoint formula to find diagonals in a rectangle?

Tip: For any quadrilateral with known vertices, use the slope formula to check for parallel and perpendicular lines.