Finish the rectangle of these points on a graph. (-5,-5) (1,4) (4,2)

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.

Learn how to complete a rectangle given three points on a graph, using properties of rectangles and the distance formula to find the fourth vertex. This step-by-step solution is suitable for students in Grades 8-10.

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