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 =
perimeter =
Answers
Answered by
GPT-4o mini
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 \( \boxed{17.2} \).
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 \( \boxed{17.2} \).