Question

Determine the coordinates of rectangle ABCD. use the coordinates to compute the area of the rectangle
A(-3,-2) B(4,-2) C(4,1) D(-3,1)

____ units^2
GPT-4o mini
To determine the area of rectangle ABCD using its coordinates, we first identify the vertices:

- A(-3, -2)
- B(4, -2)
- C(4, 1)
- D(-3, 1)

Next, we need to find the lengths of the sides of the rectangle.

1. **Length of AB (or CD)**:
Since points A and B have the same y-coordinate (-2), the distance between them is calculated as follows:
\[
AB = x_B - x_A = 4 - (-3) = 4 + 3 = 7 \text{ units}
\]

2. **Length of AD (or BC)**:
Since points A and D have the same x-coordinate (-3), the distance between them is calculated as follows:
\[
AD = y_D - y_A = 1 - (-2) = 1 + 2 = 3 \text{ units}
\]

Now that we have the lengths of the sides of the rectangle, we can compute the area \( A \) using the formula:
\[
\text{Area} = \text{Length} \times \text{Width}
\]
Substituting the lengths we found:
\[
\text{Area} = AB \times AD = 7 \times 3 = 21 \text{ units}^2
\]

Thus, the area of rectangle ABCD is \( \text{21 units}^2 \).
use the coordinates to compute the approximate area of the rectangle. Round your answer to the nearest whole number.
A(4,2) B(10,4) C(6,16) D(0,14)


______ units^2
GPT-4o mini
To compute the area of the shape formed by points A(4,2), B(10,4), C(6,16), and D(0,14), we first need to verify whether these points form a rectangle. However, from the coordinates provided, it appears that they are not the vertices of a rectangle.

Instead, we can use the shoelace formula (also known as Gauss's area formula) to calculate the area of a polygon given its vertices in order.

**Coordinates of vertices in order:**
- A(4, 2)
- B(10, 4)
- C(6, 16)
- D(0, 14)

The shoelace formula is:

\[
\text{Area} = \frac{1}{2} \left| \sum (x_iy_{i+1} - y_ix_{i+1}) \right|
\]

Setting up the coordinates:
\[
\begin{array}{c|c}
x & y \\
\hline
4 & 2 \\
10 & 4 \\
6 & 16 \\
0 & 14 \\
4 & 2 \\
\end{array}
\]

Calculating the sum:

### Step 1: Calculate \( x_iy_{i+1} \)
- From A to B: \( 4 \cdot 4 = 16 \)
- From B to C: \( 10 \cdot 16 = 160 \)
- From C to D: \( 6 \cdot 14 = 84 \)
- From D to A: \( 0 \cdot 2 = 0 \)

Sum of \( x_iy_{i+1} = 16 + 160 + 84 + 0 = 260 \)

### Step 2: Calculate \( y_ix_{i+1} \)
- From A to B: \( 2 \cdot 10 = 20 \)
- From B to C: \( 4 \cdot 6 = 24 \)
- From C to D: \( 16 \cdot 0 = 0 \)
- From D to A: \( 14 \cdot 4 = 56 \)

Sum of \( y_ix_{i+1} = 20 + 24 + 0 + 56 = 100 \)

### Step 3: Calculate the area
Using the shoelace formula:
\[
\text{Area} = \frac{1}{2} \left| 260 - 100 \right| = \frac{1}{2} \left| 160 \right| = \frac{160}{2} = 80 \text{ units}^2
\]

Rounding this to the nearest whole number, the area of the polygon formed by points A, B, C, and D is:

**80 units²**.
Fiona draws a triangle on a coordinate grid with vertices at (6,2), (4,6), and (1,2). What is the area in square units of Fiona's triangle?
GPT-4o mini
To find the area of a triangle given its vertices, we can use the formula:

\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Given the vertices of the triangle:
- \( A(6, 2) \)
- \( B(4, 6) \)
- \( C(1, 2) \)

Here, we assign:
- \( (x_1, y_1) = (6, 2) \)
- \( (x_2, y_2) = (4, 6) \)
- \( (x_3, y_3) = (1, 2) \)

Now we can substitute the values into the formula:

\[
\text{Area} = \frac{1}{2} \left| 6(6 - 2) + 4(2 - 2) + 1(2 - 6) \right|
\]

Calculating step-by-step:

1. Calculate \( 6(6 - 2) = 6(4) = 24 \)
2. Calculate \( 4(2 - 2) = 4(0) = 0 \)
3. Calculate \( 1(2 - 6) = 1(-4) = -4 \)

Now plug these results back into the formula:

\[
\text{Area} = \frac{1}{2} \left| 24 + 0 - 4 \right| = \frac{1}{2} \left| 20 \right| = \frac{20}{2} = 10
\]

Therefore, the area of Fiona's triangle is

\[
\boxed{10} \text{ square units}.
\]