If the coordinates of the endpoints of a diagonal of square are `(-2, -3) and (5, 4)`, what is the number of square units in the area of the square?

Answer

To find the area of the square given the coordinates of the endpoints of its diagonal, we can follow these steps:

### Step 1: Identify the coordinates of the endpoints
The endpoints of the diagonal of the square are given as:
- Point A: (-2, -3)
- Point B: (5, 4)

### Step 2: Use the distance formula to find the length of the diagonal
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B:
\[ d = \sqrt{(5 - (-2))^2 + (4 - (-3))^2} \]
\[ = \sqrt{(5 + 2)^2 + (4 + 3)^2} \]
\[ = \sqrt{(7)^2 + (7)^2} \]
\[ = \sqrt{49 + 49} \]
\[ = \sqrt{98} \]
\[ = 7\sqrt{2} \]

### Step 3: Relate the diagonal to the side of the square
For a square, the relationship between the length of the diagonal \(d\) and the side length \(s\) is given by:
\[ d = s\sqrt{2} \]
Setting this equal to the diagonal we found:
\[ 7\sqrt{2} = s\sqrt{2} \]

### Step 4: Solve for the side length \(s\)
Dividing both sides by \(\sqrt{2}\):
\[ s = 7 \]

### Step 5: Calculate the area of the square
The area \(A\) of a square is given by:
\[ A = s^2 \]
Substituting the value of \(s\):
\[ A = 7^2 = 49 \]

### Final Answer
The area of the square is \(49\) square units.

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