To find the length of the diagonal of the square defined by the points (1,2), (3,2), (3,4), and (1,4), we can use the Pythagorean Theorem.
First, identify the length of one side of the square. The points (1,2) and (3,2) give us the horizontal length:
\[ \text{Length of the side} = x_2 - x_1 = 3 - 1 = 2 \]
Now, we can find the diagonal of the square using the Pythagorean Theorem, which states:
\[ c = \sqrt{a^2 + b^2} \]
where \( c \) is the length of the diagonal, and \( a \) and \( b \) are the lengths of the sides of the square. Here, since it’s a square, both \( a \) and \( b \) will be equal to the side length, which is 2. Thus:
\[ c = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
Calculating \( 2\sqrt{2} \):
\[ 2\sqrt{2} \approx 2 \times 1.414 = 2.828 \]
Rounding to the nearest hundredth, the length of the diagonal is approximately \( 2.83 \).
So the answer is:
3. 2.83