Question

A rectangle in the coordinate plane has vertices at (5,6), (9,2), and (5,2). What are the coordinates of the fourth vertex?%0D%0A(6,5)%0D%0A(2,5)%0D%0A(2,9)%0D%0A(9,6)%0D%0ASubmit

Answer 1

To find the coordinates of the fourth vertex of the rectangle, we can use the properties of rectangles in the coordinate plane.

Given the three vertices:
1. \( A(5,6) \)
2. \( B(9,2) \)
3. \( C(5,2) \)

We can determine which pairs of points share the same x-coordinate or y-coordinate:

- Points \( A(5,6) \) and \( C(5,2) \) have the same x-coordinate (5).
- Points \( B(9,2) \) and \( C(5,2) \) have the same y-coordinate (2).

Now, we can find the x- and y-coordinates of the fourth vertex \( D \):

1. The fourth vertex \( D \) must share the x-coordinate with \( B \) since it needs to be in line vertically with \( B \). Therefore, the x-coordinate of \( D \) will be \( 9 \).
2. The fourth vertex \( D \) must share the y-coordinate with \( A \) since it needs to be in line horizontally with \( A \). Therefore, the y-coordinate of \( D \) will be \( 6 \).

Thus, the coordinates of the fourth vertex \( D \) are \( (9, 6) \).

Therefore, the completion of the rectangle is:

**The coordinates of the fourth vertex are (9, 6).**

However, please note that (9, 6) is not one of the provided options. If we instead analyze our choices, we realize that there was a miscalculation in identifying points.

Since we already know points A, B, and C, we try to pair them as follows:

Using the shared coordinates:
1. From C (5,2), we know that the fourth vertex D should share the x with A and y with B.

So:

- The fourth vertex should be \( D = (9, 6) \),
which is not one of the options you provided.
Double-check the points given or opt for the physical drawing if possible next time.

And still, other options are:

- (6,5) --- does not connect the rectangle
- (2,5) --- does not connect the rectangle
- (2,9) --- does not connect the rectangle
- (9,6) --- correct vertex found

So, based on correct calculations, we would expect a full verification if any interaction is present or mismatched options slightly derived.

Let me know how else you can assist!