Question

Use the image to answer the 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

2.83
2.83

4
4

4.24
4.24

8

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 use the Pythagorean Theorem.

First, we can determine the width and height of the rectangle:

- Width (from A to B or D to C):
\[
x_B - x_A = 3 - 1 = 2
\]

- Height (from A to D or B to C):
\[
y_C - y_B = 4 - 2 = 2
\]

Now, we can apply the Pythagorean Theorem, which states:
\[
c = \sqrt{a^2 + b^2}
\]
where \(a\) and \(b\) are the sides of the rectangle, and \(c\) is the diagonal.

Substituting the width and height into the equation:
\[
c = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

Now we can approximate \(2\sqrt{2}\):
\[
\sqrt{2} \approx 1.414 \Rightarrow 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828
\]

Rounding this to the nearest hundredth gives us approximately \(2.83\).

Thus, the length of the diagonal rounded to the nearest hundredth is:

**2.83**