To find the length of the diagonal of the rectangle formed by the points A, B, C, and D, we can use the Pythagorean Theorem.
The diagonal of the rectangle can be found by determining the lengths of its sides. From the points provided:
- Point A: (1, 2)
- Point B: (3, 2)
- Point C: (3, 4)
- Point D: (1, 4)
Step 1: Find the lengths of the sides of the rectangle.
The length of the horizontal side (AB or CD):
- The x-coordinates of A and B are 1 and 3 respectively.
- Length AB = x-coordinate of B - x-coordinate of A = 3 - 1 = 2.
The length of the vertical side (AD or BC):
- The y-coordinates of A and D are 2 and 4 respectively.
- Length AD = y-coordinate of D - y-coordinate of A = 4 - 2 = 2.
Step 2: Use the Pythagorean Theorem to find the diagonal (d).
According to the Pythagorean Theorem: \[ d^2 = a^2 + b^2 \] where \( a \) is the length of the horizontal side, and \( b \) is the length of the vertical side.
In this instance, both \( a \) and \( b \) are equal to 2:
- \( a = 2 \)
- \( b = 2 \)
So, \[ d^2 = 2^2 + 2^2 = 4 + 4 = 8 \]
Now, take the square root to find \( d \): \[ d = \sqrt{8} = \sqrt{4 \times 2} = 2 \sqrt{2} \approx 2.828427 \]
Step 3: Round to the nearest hundredth.
So the length of the diagonal, rounded to the nearest hundredth, is: \[ d \approx 2.83 \]
Thus, the length of the diagonal is approximately 2.83.