Question

Use the coordinates to compute the exact area of the rectangle. A=0,0 B=4,-4 C=7,-1 D=3,3

Answer 1

To find the area of the rectangle defined by the coordinates A(0,0), B(4,-4), C(7,-1), and D(3,3), we first need to confirm whether these points indeed form a rectangle, and then we can calculate the area.

### Step 1: Verify Coordinates Form a Rectangle

1. **Identify Vertices:**
- A(0, 0)
- B(4, -4)
- C(7, -1)
- D(3, 3)

2. **Check Distances to Determine Perpendicular Sides:**
- Calculate lengths of the sides:
- AB = √[(4-0)² + (-4-0)²] = √[4² + (-4)²] = √[16 + 16] = √32 = 4√2
- BC = √[(7-4)² + (-1+4)²] = √[3² + (3)²] = √[9 + 9] = √18 = 3√2
- CD = √[(7-3)² + (-1-3)²] = √[4² + (-4)²] = √[16 + 16] = √32 = 4√2
- DA = √[(3-0)² + (3-0)²] = √[3² + 3²] = √[9 + 9] = √18 = 3√2

3. **Check Angles to Verify Right Angles:**
- Using the slope formula, we find:
- Slope of AB = (-4-0)/(4-0) = -1
- Slope of BC = (-1-(-4))/(7-4) = 3/3 = 1
- Slope of CD = (3-(-1))/(3-7) = 4/(-4) = -1
- Slope of DA = (3-0)/(3-0) = 1

Since opposite sides are equal and slopes of consecutive sides are negative reciprocals, we conclude these points form a rectangle.

### Step 2: Calculate the Area

1. **Dimensions of the Rectangle:**
- Length (along AB or CD) = 4√2
- Width (along BC or DA) = 3√2

2. **Area of the Rectangle:**
\[
\text{Area} = \text{Length} \times \text{Width} = (4\sqrt{2}) \times (3\sqrt{2})
\]
\[
\text{Area} = 12 \times 2 = 24
\]

### Conclusion

Hence, the exact area of the rectangle is \(\boxed{24}\) square units.