Question
To the nearest tenth, what is the area of the rectangle ABCD with vertices A(-3,-2), B(-5,2), C(3,6),and D(5,2)?
26.0
1 460.0
32.0
40
26.0
1 460.0
32.0
40
Answers
GPT-4o mini
To find the area of rectangle ABCD with vertices \( A(-3,-2) \), \( B(-5,2) \), \( C(3,6) \), and \( D(5,2) \), we first need to determine the lengths of its sides.
We can calculate the lengths of two adjacent sides using the distance formula, which is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Let's find the length of side AB:
- \( A(-3, -2) \) and \( B(-5, 2) \):
\[
AB = \sqrt{((-5) - (-3))^2 + (2 - (-2))^2}
\]
\[
= \sqrt{(-5 + 3)^2 + (2 + 2)^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
Next, we calculate the length of side AD:
- \( A(-3, -2) \) and \( D(5, 2) \):
\[
AD = \sqrt{((5) - (-3))^2 + (2 - (-2))^2}
\]
\[
= \sqrt{(5 + 3)^2 + (2 + 2)^2} = \sqrt{(8)^2 + (4)^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}
\]
Now we can find the area of rectangle ABCD using the formula for the area \( A = \text{length} \times \text{width} \):
\[
\text{Area} = AB \times AD = (2\sqrt{5}) \times (4\sqrt{5}) = 8 \times 5 = 40
\]
Thus, the area of rectangle ABCD is \( 40 \).
To the nearest tenth, the area is:
\[
\boxed{40.0}
\]
We can calculate the lengths of two adjacent sides using the distance formula, which is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Let's find the length of side AB:
- \( A(-3, -2) \) and \( B(-5, 2) \):
\[
AB = \sqrt{((-5) - (-3))^2 + (2 - (-2))^2}
\]
\[
= \sqrt{(-5 + 3)^2 + (2 + 2)^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
Next, we calculate the length of side AD:
- \( A(-3, -2) \) and \( D(5, 2) \):
\[
AD = \sqrt{((5) - (-3))^2 + (2 - (-2))^2}
\]
\[
= \sqrt{(5 + 3)^2 + (2 + 2)^2} = \sqrt{(8)^2 + (4)^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}
\]
Now we can find the area of rectangle ABCD using the formula for the area \( A = \text{length} \times \text{width} \):
\[
\text{Area} = AB \times AD = (2\sqrt{5}) \times (4\sqrt{5}) = 8 \times 5 = 40
\]
Thus, the area of rectangle ABCD is \( 40 \).
To the nearest tenth, the area is:
\[
\boxed{40.0}
\]
Are you sure it's right
GPT-4o mini
Let's go through the calculations carefully to ensure the correctness of the area of rectangle \( ABCD \) with the vertices \( A(-3,-2) \), \( B(-5,2) \), \( C(3,6) \), and \( D(5,2) \).
### Step 1: Identify the Rectangle's Vertices
The given vertices are:
- \( A(-3, -2) \)
- \( B(-5, 2) \)
- \( C(3, 6) \)
- \( D(5, 2) \)
### Step 2: Verify the Opposite Sides
To check if these points indeed form a rectangle, we need to determine the lengths of the sides and verify if opposite sides are equal:
### Length of AB
Using the distance formula:
\[
AB = \sqrt{((-5) - (-3))^2 + (2 - (-2))^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
### Length of BC
\[
BC = \sqrt{((3) - (-5))^2 + (6 - 2)^2} = \sqrt{(8)^2 + (4)^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}
\]
### Length of CD
\[
CD = \sqrt{((5) - (3))^2 + (2 - 6)^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
### Length of DA
\[
DA = \sqrt{((-3) - (5))^2 + ((-2) - (2))^2} = \sqrt{(-8)^2 + (-4)^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}
\]
### Step 3: Compare Sides
From the above calculations:
- \( AB = 2\sqrt{5} \) (length)
- \( BC = 4\sqrt{5} \) (width)
- \( CD = 2\sqrt{5} \) (length)
- \( DA = 4\sqrt{5} \) (width)
We see that opposite sides are equal, confirming this shape is indeed a rectangle.
### Step 4: Calculate the Area
To find the area \( A \):
\[
\text{Area} = AB \times BC = (2\sqrt{5}) \times (4\sqrt{5}) = 8 \times 5 = 40
\]
### Conclusion
The calculations confirm that the area of rectangle \( ABCD \) is indeed \( 40.0 \). The final answer is:
\[
\boxed{40.0}
\]
This conclusion is consistent with the dimensions calculated, and thus the result is verified to be correct.
### Step 1: Identify the Rectangle's Vertices
The given vertices are:
- \( A(-3, -2) \)
- \( B(-5, 2) \)
- \( C(3, 6) \)
- \( D(5, 2) \)
### Step 2: Verify the Opposite Sides
To check if these points indeed form a rectangle, we need to determine the lengths of the sides and verify if opposite sides are equal:
### Length of AB
Using the distance formula:
\[
AB = \sqrt{((-5) - (-3))^2 + (2 - (-2))^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
### Length of BC
\[
BC = \sqrt{((3) - (-5))^2 + (6 - 2)^2} = \sqrt{(8)^2 + (4)^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}
\]
### Length of CD
\[
CD = \sqrt{((5) - (3))^2 + (2 - 6)^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
### Length of DA
\[
DA = \sqrt{((-3) - (5))^2 + ((-2) - (2))^2} = \sqrt{(-8)^2 + (-4)^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}
\]
### Step 3: Compare Sides
From the above calculations:
- \( AB = 2\sqrt{5} \) (length)
- \( BC = 4\sqrt{5} \) (width)
- \( CD = 2\sqrt{5} \) (length)
- \( DA = 4\sqrt{5} \) (width)
We see that opposite sides are equal, confirming this shape is indeed a rectangle.
### Step 4: Calculate the Area
To find the area \( A \):
\[
\text{Area} = AB \times BC = (2\sqrt{5}) \times (4\sqrt{5}) = 8 \times 5 = 40
\]
### Conclusion
The calculations confirm that the area of rectangle \( ABCD \) is indeed \( 40.0 \). The final answer is:
\[
\boxed{40.0}
\]
This conclusion is consistent with the dimensions calculated, and thus the result is verified to be correct.