The vertices of a reactangle ABCD are `A (-1, 0) B (2,0), C (a,b) and D(-1, 4).` Then the length of the diagonal AC is

Answer

To find the length of the diagonal AC in the rectangle ABCD with given vertices A(-1, 0), B(2, 0), C(a, b), and D(-1, 4), we can follow these steps:

### Step 1: Identify the coordinates of points A and D
The coordinates of point A are given as A(-1, 0) and point D as D(-1, 4).

### Step 2: Determine the coordinates of point C
Since ABCD is a rectangle, the opposite sides are equal and parallel. The x-coordinates of points A and D are the same (-1), so point C must have the same y-coordinate as point D (which is 4) and the same x-coordinate as point B (which is 2). Therefore, the coordinates of point C are C(2, 4).

### Step 3: Use the distance formula to find the length of diagonal AC
The distance formula between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of points A(-1, 0) and C(2, 4) into the formula:
- \( x_1 = -1, y_1 = 0 \)
- \( x_2 = 2, y_2 = 4 \)

Now, calculate the distance:
\[ d = \sqrt{(2 - (-1))^2 + (4 - 0)^2} \]
\[ = \sqrt{(2 + 1)^2 + (4)^2} \]
\[ = \sqrt{(3)^2 + (4)^2} \]
\[ = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \]

### Conclusion
The length of the diagonal AC is 5.

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