If the three vertices of a rectangle taken in order are the points (2, -2), (8, 4) and (5, 7). The coordinates of the fourth vertex is

Options

A. (1, 1)
B. (1, -1)
C. (-1, 1)
D. None of these

Answer

To find the coordinates of the fourth vertex of the rectangle given the coordinates of three vertices, we can use the property of the diagonals of a rectangle. The diagonals of a rectangle bisect each other, which means that the sum of the coordinates of one diagonal equals the sum of the coordinates of the other diagonal.

Let's denote the vertices of the rectangle as follows:
- A (2, -2)
- B (8, 4)
- C (5, 7)
- D (x, y) (the unknown vertex we need to find)

According to the property mentioned, we have:

\[ A + C = B + D \]

This can be expressed in terms of coordinates:

\[ (2, -2) + (5, 7) = (8, 4) + (x, y) \]

Now, let's break this down step by step.

### Step 1: Add the coordinates of A and C
- For the x-coordinates: \( 2 + 5 = 7 \)
- For the y-coordinates: \( -2 + 7 = 5 \)

So, we have:

\[ (7, 5) \]

### Step 2: Set up the equation for B and D
Now, we need to set the sum of the coordinates of B and D equal to the sum we just calculated:

\[ (7, 5) = (8, 4) + (x, y) \]

### Step 3: Break this down into two equations
From the above equation, we can derive two separate equations:

1. For the x-coordinates:
\[ 7 = 8 + x \]

2. For the y-coordinates:
\[ 5 = 4 + y \]

### Step 4: Solve for x and y
Now, let's solve each equation:

1. From \( 7 = 8 + x \):
\[ x = 7 - 8 \]
\[ x = -1 \]

2. From \( 5 = 4 + y \):
\[ y = 5 - 4 \]
\[ y = 1 \]

### Conclusion
Thus, the coordinates of the fourth vertex D are:

\[ D (-1, 1) \]

### Final Answer
The coordinates of the fourth vertex are (-1, 1).

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