(4, 5)'(9, 5) and (5,1) are three of a quadrilateral whose opposite sides are equal and parallel.there are three possibilities for the co-ordinates of the missing vertex.Find all three,and draw the three quadrilateral on a grid

To find the three possibilities for the coordinates of the missing vertex, we can use the fact that opposite sides of the quadrilateral are equal and parallel. Let's go step by step:

Step 1: Identify the given coordinates:
- (4, 5)
- (9, 5)
- (5, 1)

Step 2: Determine the missing vertex:
Since the opposite sides are equal and parallel, we know that the line segment between (4, 5) and (9, 5) is parallel to the line segment between (5, 1) and the missing vertex. Additionally, the line segment between (4, 5) and (5, 1) is parallel to the line segment between (9, 5) and the missing vertex.

Step 3: Find the possibilities for the missing vertex:
To find the possibilities, we need to look for points that are equidistant and parallel from both (4, 5) and (9, 5). Similarly, we need to find points that are equidistant and parallel from both (4, 5) and (5, 1).

One possibility for the missing vertex is the point (1, 1), as it is equidistant and parallel to both (4, 5) and (9, 5), as well as (4, 5) and (5, 1).

The second possibility for the missing vertex is the point (6, 1), as it also fulfills the conditions mentioned above.

The third possibility for the missing vertex is the point (8, 1), which satisfies the given requirements.

Step 4: Draw the quadrilaterals on a grid:
To visualize the three quadrilaterals, plot the given coordinates and connect them with the three possibilities for the missing vertex:

Quadrilateral 1: (4, 5), (5, 1), (9, 5), (1, 1)
Quadrilateral 2: (4, 5), (5, 1), (9, 5), (6, 1)
Quadrilateral 3: (4, 5), (5, 1), (9, 5), (8, 1)

Here is a rough illustration of the three quadrilaterals on a grid:

```
| |
| |
(1, 1)| |(8, 1)
| |
| |
| |
| |
(5, 1)| |(9, 5)
| |
| |
(4, 5)| |(6, 1)
| |
| |
```

Please note that the coordinates are roughly estimated for visualization purposes.

To find the three possibilities for the coordinates of the missing vertex, we need to determine the pattern or relationship between the given coordinates. The quadrilateral has opposite sides that are both equal and parallel. Let's start by examining the given coordinates and their properties.

The first set of coordinates given is (4, 5). Let's call this point A.
The second set of coordinates is (9, 5). Let's call this point B.
The third set of coordinates is (5, 1). Let's call this point C.

Since opposite sides are equal and parallel, we can deduce that side AB is parallel to side CD, and side BC is parallel to side AD.

Now, let's determine the possible coordinates for point D, the missing vertex.

Possibility 1:
Since side AB is parallel to side CD, the y-coordinate of point D should be the same as point C. So, one possibility for point D is (9, 1).

Possibility 2:
Since side BC is parallel to side AD, the x-coordinate of point D should be the same as point A. So, another possibility for point D is (4, 1).

Possibility 3:
To find the third possibility, we need to find the midpoint of the diagonal AC. The midpoint formula is given by: ( (x1 + x2) / 2 , (y1 + y2) / 2 ).

For points A (4, 5) and C (5, 1), the midpoint is:
( (4 + 5) / 2 , (5 + 1) / 2 ) = (4.5, 3)

So, the third possibility for point D is (4.5, 3).

Now, let's draw the three quadrilaterals on a grid.

First Quadrilateral:
A (4, 5), B (9, 5), C (5, 1), D (9, 1)

Second Quadrilateral:
A (4, 5), B (9, 5), C (5, 1), D (4, 1)

Third Quadrilateral:
A (4, 5), B (9, 5), C (5, 1), D (4.5, 3)

Please note that the grid dimensions and scale are not mentioned in the question, so the quadrilaterals will be drawn based on a general grid and scale.