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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

4.24
4.24

4
4

2.83
2.83

8
8
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9 answers

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.

  1. First, we calculate the length of one side of the rectangle:

    • The length between points A(1, 2) and D(1, 4) is the difference in the y-coordinates: \[ 4 - 2 = 2 \]

    • The length between points A(1, 2) and B(3, 2) is the difference in the x-coordinates: \[ 3 - 1 = 2 \]

    Therefore, the sides of the rectangle are both \(2\) units.

  2. Next, to find the diagonal \(d\), we apply the Pythagorean Theorem: \[ d = \sqrt{(\text{length})^2 + (\text{width})^2} \] Substituting the lengths we found: \[ d = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \]

  3. Simplifying \(\sqrt{8}\): \[ d = \sqrt{4 \times 2} = 2\sqrt{2} \approx 2 \times 1.414 = 2.828 \]

  4. Rounding this to the nearest hundredth gives: \[ d \approx 2.83 \]

Hence, the length of the diagonal is 2.83.

Apply the Pythagorean Theorem to find the length between (2, 5)
and (7, 3)
. Round your answer to the nearest hundredth, if necessary.(1 point)
Responses

5.39
5.39

2.65
2.65

4.58
4.58

29
29

To find the distance between the points \( (2, 5) \) and \( (7, 3) \) using the Pythagorean Theorem, we can use the distance formula:

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

Here, \( (x_1, y_1) = (2, 5) \) and \( (x_2, y_2) = (7, 3) \).

  1. Calculate the differences in the coordinates:

    • \( x_2 - x_1 = 7 - 2 = 5 \)
    • \( y_2 - y_1 = 3 - 5 = -2 \)

  2. Substitute into the distance formula: \[ d = \sqrt{(5)^2 + (-2)^2} \] \[ d = \sqrt{25 + 4} = \sqrt{29} \]

  3. Calculate \( \sqrt{29} \): \[ d \approx 5.385 \]

  4. Rounding this to the nearest hundredth gives: \[ d \approx 5.39 \]

Thus, the length between the points \( (2, 5) \) and \( (7, 3) \) is 5.39.

Apply the Pythagorean Theorem to find the length between (−1, 6)
and (5, −2)
. Round your answer to the nearest hundredth, if necessary.(1 point)
Responses

14
14

100
100

9.22
9.22

10

To find the distance between the points \( (-1, 6) \) and \( (5, -2) \) using the Pythagorean Theorem, we will use the distance formula:

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

Here, \( (x_1, y_1) = (-1, 6) \) and \( (x_2, y_2) = (5, -2) \).

  1. Calculate the differences in the coordinates:

    • \( x_2 - x_1 = 5 - (-1) = 5 + 1 = 6 \)
    • \( y_2 - y_1 = -2 - 6 = -8 \)

  2. Substitute into the distance formula: \[ d = \sqrt{(6)^2 + (-8)^2} \] \[ d = \sqrt{36 + 64} = \sqrt{100} \]

  3. Calculate \( \sqrt{100} \): \[ d = 10 \]

Therefore, the length between the points \( (-1, 6) \) and \( (5, -2) \) is 10.

Use the image to answer the question.

A graph shows four quadrants with the x-axis and y-axis both ranging from negative 5 to positive 5 in increments of 1. Unlabeled data points marked are at left parenthesis negative 2 comma negative 1 right parenthesis, and at left parenthesis 2 comma 4 right parenthesis. A straight line joins the two lines with a positive slope.

Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.

(1 point)
Responses

41
41

5.57
5.57

6:40
6:40

9
9
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To find the distance between the points \( (-2, -1) \) and \( (2, 4) \) using the Pythagorean Theorem (or the distance formula), we use the formula:

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

Where:

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

  1. Calculate the differences in the coordinates:

    • \( x_2 - x_1 = 2 - (-2) = 2 + 2 = 4 \)
    • \( y_2 - y_1 = 4 - (-1) = 4 + 1 = 5 \)

  2. Substitute into the distance formula: \[ d = \sqrt{(4)^2 + (5)^2} \] \[ d = \sqrt{16 + 25} = \sqrt{41} \]

  3. Calculate \( \sqrt{41} \): \[ d \approx 6.403 \]

  4. Rounding this to the nearest hundredth gives: \[ d \approx 6.40 \]

Thus, the length between the points \( (-2, -1) \) and \( (2, 4) \) is 6.40.

Use the image to answer the question.

A graph shows four quadrants with the x-axis and y-axis both ranging from negative 5 to positive 5 in increments of 1. Data points marked are point A at left parenthesis 2 comma negative 3 right parenthesis, and point B at left parenthesis 5 comma 5 right parenthesis.

Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.

(1 point)
Responses

73
73

8.54
8.54

11
11

7.62
7.62
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To find the distance between the points \( A(2, -3) \) and \( B(5, 5) \) using the Pythagorean Theorem (or the distance formula), we can use the formula:

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

Where:

  • \( (x_1, y_1) = (2, -3) \)
  • \( (x_2, y_2) = (5, 5) \)

  1. Calculate the differences in the coordinates:

    • \( x_2 - x_1 = 5 - 2 = 3 \)
    • \( y_2 - y_1 = 5 - (-3) = 5 + 3 = 8 \)

  2. Substitute into the distance formula: \[ d = \sqrt{(3)^2 + (8)^2} \] \[ d = \sqrt{9 + 64} = \sqrt{73} \]

  3. Calculate \( \sqrt{73} \): \[ d \approx 8.544 \]

  4. Rounding this to the nearest hundredth gives: \[ d \approx 8.54 \]

Thus, the length between the points \( A(2, -3) \) and \( B(5, 5) \) is 8.54.