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

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)
perimeter =

GPT-4o mini · Answer

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.

Coordinates of A and B:
- A = (-1, -1)
- B = (1, 2)
Finding the length of diagonal BD: The length of diagonal BD is given as 6 units.
Finding coordinates of D: Let the coordinates of point D be $(x_{D}, y_{D})$ .
Using the distance formula: The distance between points B and D is given by: $B D = \sqrt{(x_{D} - 1)^{2} + (y_{D} - 2)^{2}} = 6$

Squaring both sides, we obtain: $(x_{D} - 1)^{2} + (y_{D} - 2)^{2} = 36$
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: $C = (- 1, 2)$
Finding coordinates of D: Point D will have the same x-coordinate as point B and the same y-coordinate as point A: $D = (1, - 1)$
Double-checking the distance BD: Now, we check the distance BD: $B D = \sqrt{(1 - 1)^{2} + (- 1 - 2)^{2}} = \sqrt{0 + (- 3)^{2}} = \sqrt{9} = 3$

It seems we need to re-evaluate since it must equal 6 units.
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: $d i s t a n c e = \sqrt{(x_{D} - 1)^{2} + (y_{D} - 2)^{2}} = 6$

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: $| y_{D} - 2 | = 6 ⟹ y_{D} = 8 or y_{D} = - 4$ Let's choose $D = (1, - 4)$ .
Calculating dimensions of rectangle:
- Length (vertical dimension) = distance from A to D: $Length = 4 - (- 1) = 5$
- Width (horizontal dimension) = distance from A to B: $Width = \sqrt{(1 - (- 1))^{2} + (2 - (- 1))^{2}} = \sqrt{(2)^{2} + (3)^{2}} = \sqrt{4 + 9} = \sqrt{13}$
Perimeter calculation: $Perimeter = 2 \times (Length + Width) = 2 \times (5 + \sqrt{13})$
Finding numerical approximation: $\sqrt{13} \approx 3.6 ⟹ 5 + \sqrt{13} \approx 8.6$ Thus: $Perimeter \approx 2 \times 8.6 \approx 17.2$

So the perimeter of rectangle ABCD is approximately $17.2$ .