A field is on a slant; one corner of its slant is equal to 3 and is at (3, 0). The other corner of that side is equal to 4 and is at (0, 4). The other 2 points of the square are at (4, 7) and (7, 3). Using the Pythagorean Theorem, what is the area of the field?

Answer 1

To find the area of the field defined by the four points, we first need to determine the lengths of its sides using the distance formula. The given points of the field are:

• A(3, 0)
• B(0, 4)
• C(4, 7)
• D(7, 3)

Next, we will calculate the distances between adjacent points to confirm that the shape looks like a square, and if so, find the area.

Step 1: Distance between A and B

Using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] we calculate the distance \( AB \) as follows: \[ AB = \sqrt{(0 - 3)^2 + (4 - 0)^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \]

Step 2: Distance between B and C

Now we calculate the distance \( BC \): \[ BC = \sqrt{(4 - 0)^2 + (7 - 4)^2} = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \]

Step 3: Distance between C and D

Next, we calculate the distance \( CD \): \[ CD = \sqrt{(7 - 4)^2 + (3 - 7)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \]

Step 4: Distance between D and A

Finally, we calculate the distance \( DA \): \[ DA = \sqrt{(3 - 7)^2 + (0 - 3)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \]

Step 5: Area Calculation

Since all sides are equal (5 units), and we have a quadrilateral with sides of equal length, we need to ensure it is a square. To confirm that it is a square, we can check the slopes of the sides (the slopes of adjacent sides being negative reciprocals indicates a right angle).

Calculating the slopes:

• Slope of \( AB \): \[ \text{slope of } AB = \frac{4 - 0}{0 - 3} = -\frac{4}{3} \]
• Slope of \( BC \): \[ \text{slope of } BC = \frac{7 - 4}{4 - 0} = \frac{3}{4} \]
• Slope of \( CD \): \[ \text{slope of } CD = \frac{3 - 7}{7 - 4} = -\frac{4}{3} \]
• Slope of \( DA \): \[ \text{slope of } DA = \frac{0 - 3}{3 - 7} = \frac{3}{4} \]

The products of the slopes of adjacent sides \( (AB \) and \( BC, \) \( BC \) and \( CD, \) \( CD \) and \( DA, \) and \( DA \) and \( AB) \) all yield \(-1\), confirming that we have right angles.

Step 6: Area

Thus, since this quadrilateral is a square with each side measuring 5 units, the area \( A \) is: \[ A = \text{side}^2 = 5^2 = 25. \]

Therefore, the area of the field is \( \boxed{25} \).