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.

Question

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

8
8

4
4

4.24

Answer 1

To find the length of the diagonal of the rectangle formed by points A, B, C, and D, we can use the Pythagorean Theorem. The rectangle has the following vertices:

• A(1, 2)
• B(3, 2)
• C(3, 4)
• D(1, 4)

We can find the length of the diagonal by using the coordinates of any two opposite corners, such as points A and C.

The distance (d) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here we will use points A (1, 2) and C (3, 4):

• \(x_1 = 1\), \(y_1 = 2\)
• \(x_2 = 3\), \(y_2 = 4\)

Plugging in these values:

\[ d = \sqrt{(3 - 1)^2 + (4 - 2)^2} \]

Calculating this step-by-step:

1. Calculate the differences:

   • \(3 - 1 = 2\)
   • \(4 - 2 = 2\)

2. Square the differences:

   • \(2^2 = 4\)
   • \(2^2 = 4\)

3. Add the squares:

   • \(4 + 4 = 8\)

4. Take the square root:

   • \(d = \sqrt{8} = 2\sqrt{2} \approx 2.83\)

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

The correct response is:

2.83