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Question

A graph shows quadrant 1 with the x-axis and y-axis ranging from 0 to positive 5. Data points marked are point A at left parenthesis 1 comma 2 right parenthesis, point B at left parenthesis 3 comma 2 right parenthesis, point C at left parenthesis 3 comma 4 right parenthesis, and point D at left parenthesis 1 comma 4 right parenthesis. A line joins points A to B, B to C, C to D, and D to A, forming a rectangle.

Apply the Pythagorean Theorem to find the length of the diagonal of the square. Round your answer to the nearest hundredth, if necessary.

(1 point)
Responses

4.24
4.24

8
8

4
4

2.83

Answer 1

To find the length of the diagonal of the rectangle formed by the points A(1, 2), B(3, 2), C(3, 4), and D(1, 4), we can apply the Pythagorean Theorem.

First, we need to calculate the length of the sides of the rectangle:

1. Length of side AB (horizontal):

   • Points A(1, 2) and B(3, 2) have the same y-coordinate.
   • Length = x-coordinate of B - x-coordinate of A = 3 - 1 = 2.

2. Length of side AD (vertical):

   • Points A(1, 2) and D(1, 4) have the same x-coordinate.
   • Length = y-coordinate of D - y-coordinate of A = 4 - 2 = 2.

Now, to find the length of the diagonal (AC), we can use the Pythagorean Theorem, which states:

\[ c^2 = a^2 + b^2 \] where \(c\) is the length of the diagonal, and \(a\) and \(b\) are the lengths of the sides.

Here, \(a = 2\) and \(b = 2\):

\[ c^2 = 2^2 + 2^2 = 4 + 4 = 8 \] \[ c = \sqrt{8} = 2\sqrt{2} \approx 2.83 \]

So, the length of the diagonal, rounded to the nearest hundredth, is approximately 2.83.

Therefore, the correct response is:

2.83.