Question

Finding the Area of a Field
Use the image to answer the 3 questions below.
*Remember: You can use the pen or type option to write your answers.
A(4,7)
(-4, 4) (0, 1)
Field C
→ (7,3)
Field A
(0, 0)
(-4, 0)
(3, 0)
Field B
1. Count the number of squares that make one side of Field A.
Side length =
acres (worth 2 points)
2. Count the number of squares that make one side of Field B.
Side length =
acres (worth 2 points)
3. Use the Pythagorean Theorem to find the area of Field C in acres:
Area: a2 ÷ b2 = c?
Hint: You can either square your side lengths when simplifying a2 and b2 or you can count the squares in each of those fields for the same value.
(0, -3) .
1 acre =
• (3, -3) 4.840 sq yards
Area of Field C =
acres (worth 4 points)

Answer 1

Finding the Area of a Field
Use the image to answer the 3 questions below.
*Remember: You can use the pen or type option to write your answers.
A(4,7)
(-4, 4) (0, 1)
Field C
→ (7,3)
Field A
(0, 0)
(-4, 0)
(-4, 4)
Field B
1. Count the number of squares that make one side of Field A.
Side length =
acres (worth 2 points)
2. Count the number of squares that make one side of Field B.
Side length =
acres (worth 2 points)
3. Use the Pythagorean Theorem to find the area of Field C in acres:
Area: a2 ÷ b2 = c?
Hint: You can either square your side lengths when simplifying a2 and b2 or you can count the squares in each of those fields for the same value.
(0, -3) .
1 acre =
• (3, -3) 4.840 sq yards
Area of Field C =
acres

Answer 2

To answer these questions about the fields, I'll need to identify the coordinates given to calculate the side lengths and areas accordingly.

### 1. Side length of Field A
The coordinates given for Field A are (0, 0), (-4, 0), and (3, 0).
- The horizontal distance between the points (-4, 0) and (3, 0) gives us the side length:
- From -4 to 3, the distance is 3 - (-4) = 3 + 4 = 7 units.

Therefore, the side length of Field A is **7 acres**.

### 2. Side length of Field B
The coordinates given for Field B are (0, 1), (-4, 4), and (7, 3).
- We can determine the side length by measuring the vertical or horizontal distance in relation to any two of the coordinates.
- For example, measuring the horizontal distance from (-4, 4) to (0, 1):
- The relevant horizontal distance is essentially determining the x-coordinates when the y-coordinates are equal. However, since we can see these do not form a straight line for simple measurement, we can square the lengths calculated from each distance.

I'll choose (-4, 4) to (7, 3):
- Horizontal distance: 7 - (-4) = 11.
- Vertical distance: 4 - 3 = 1.

Using the distance formula to complete the dimensions:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
= \sqrt{(7 - (-4))^2 + (3 - 4)^2}
= \sqrt{(11)^2 + (-1)^2}
= \sqrt{121 + 1}
= \sqrt{122}
= 11.045 approximately.
\]
Thus, the side length of Field B is approximately **11.045 acres**.

### 3. Area of Field C using the Pythagorean Theorem
Coordinates for Field C are A(4, 7), (0, -3), and (3, -3).
- We find two sides of the triangle:
- Base from (0, -3) to (3, -3) => they are on the same horizontal level.
- Distance = 3 - 0 = 3 units.
- Height from (0, -3) to (4, 7) => determining the vertical rise.
- The vertical distance = 7 - (-3) = 7 + 3 = 10 units.

Now, we use these lengths with the Pythagorean Theorem \(a^2 + b^2 = c^2\):
- \(a = 3\), \(b = 10\):
\[
c^2 = 3^2 + 10^2 = 9 + 100 = 109.
\]
To find the area of Field C in acres, you can use the area formula for a right triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
= \frac{1}{2} \times 3 \times 10
= 15\, \text{acres}.
\]

So, the area of Field C is **15 acres**.

**Summary of Answers:**
1. Field A: **7 acres**
2. Field B: **11.045 acres**
3. Field C: **15 acres**