In rectangle ABCD, point A is at (-1,-1) and point B is at (1,2). A diagonal is drawn from point B to point D and it measures 6 units long. Using this information determine the perimeter of the rectangle. Round your answer to the nearest tenth. (1 point)

Question

perimeter =

Answer 1

To find the perimeter of rectangle ABCD, we need to determine the coordinates of points C and D, given the coordinates of points A and B.

1. Coordinates of A and B:

   • A = (-1, -1)
   • B = (1, 2)

2. Finding the length of diagonal BD: The length of diagonal BD is given as 6 units.

3. Finding coordinates of D: Let the coordinates of point D be \((x_D, y_D)\).

4. Using the distance formula: The distance between points B and D is given by: \[ BD = \sqrt{(x_D - 1)^2 + (y_D - 2)^2} = 6 \]

   Squaring both sides, we obtain: \[ (x_D - 1)^2 + (y_D - 2)^2 = 36 \]

5. Finding coordinates of C: We can also use the property of rectangles: the coordinates of point C will be vertically aligned with point B and horizontally aligned with point A. Thus, the coordinates of point C will be the same x-coordinate as A and the same y-coordinate as B: \[ C = (-1, 2) \]

6. Finding coordinates of D: Point D will have the same x-coordinate as point B and the same y-coordinate as point A: \[ D = (1, -1) \]

7. Double-checking the distance BD: Now, we check the distance BD: \[ BD = \sqrt{(1 - 1)^2 + (-1 - 2)^2} = \sqrt{0 + (-3)^2} = \sqrt{9} = 3 \]

   It seems we need to re-evaluate since it must equal 6 units.

8. Finding D with the correct size: Since distance BD = 6, the coordinates of D should be further away vertically and horizontally.

   Substituting possible values and solving the diagonal constraint: \[ distance = \sqrt{(x_D - 1)^2 + (y_D - 2)^2} = 6 \]

   Alternatively, we can define D's coordinates: Let \(D = (1, y_D)\) since x_D is fixed: \[ \sqrt{(1-1)^2 + (y_D - 2)^2} = 6 \] Therefore: \[ |y_D - 2| = 6 \implies y_D = 8 \text{ or } y_D = -4 \] Let's choose \(D = (1, -4)\).

9. Calculating dimensions of rectangle:

   • Length (vertical dimension) = distance from A to D: \[\text{Length} = 4 - (-1) = 5\]

   • Width (horizontal dimension) = distance from A to B: \[\text{Width}=\sqrt{(1 - (-1))^2 + (2 - (-1))^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13}\]

10. Perimeter calculation: \[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) = 2 \times (5 + \sqrt{13}) \]

11. Finding numerical approximation: \[ \sqrt{13} \approx 3.6 \implies 5 + \sqrt{13} \approx 8.6 \] Thus: \[ \text{Perimeter} \approx 2 \times 8.6 \approx 17.2 \]

So the perimeter of rectangle ABCD is approximately \( \boxed{17.2} \).