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Use this worksheet to record answers to the four activities that make up the Pythagorean Theorem Portfolio. When you are finished, save this worksheet with your answers and submit it for a portfolio grade.

Finding the Area of a Field
Use the image to answer to complete the activity.
(4, 7)
(4, 4) (0,4)
Field C
(7,3)
Field A
(0, 0)
(-4, 0)
(3, 0)
Field B
(0, -3)
1 acre =
→ (3, -3) 4,840 sq yards
1. Count the number of squares inside Field A. This is the area of Square A in acres. Write it below:
Area = acres
2. Count the number of squares inside Field B. This is the area of Square B in acres. Write it below:
Area = acres
3. Use the Pythagorean Theorem to find the area of Field C in acres:
Area = ________ acres

3 answers

To calculate the area for Fields A, B, and C based on the coordinates provided:

  1. Finding the Area of Field A:

    • The points given for Field A are (0, 0), (3, 0), (7, 3), and (4, 7).
    • To find the area, we can plot the points and see that it forms a triangle.
    • The base can be between (0, 0) and (3, 0), which has a length of 3.
    • The height can be determined by the vertical distance from the point (4, 7) down to the line y = 0 (the x-axis), which is 7 units.
    • The area of the triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 7 = 10.5 \text{ square units.} \]
    • To convert this into acres (given 1 acre = 4,840 square yards), we have to check how many square yards fit into the 10.5 units. If each unit is one square yard, then: \[ \text{Area in acres} = \frac{10.5}{4840} \approx 0.00217 \text{ acres.} \]
    • Thus, the Area = 0.00217 acres.

  2. Finding the Area of Field B:

    • The points for Field B are (0, 0), (0, -3), (-4, 0), and (3, 0).
    • Analyzing these points shows that the shape is another triangle.
    • The base can be between the points (0, 0) and (-4, 0) (length = 4).
    • The height is the distance from the point (0, -3) to the x-axis, which is 3 units.
    • The area calculates as: \[ \text{Area} = \frac{1}{2} \times 4 \times 3 = 6 \text{ square units.} \]
    • Converting to acres: \[ \text{Area in acres} = \frac{6}{4840} \approx 0.00124 \text{ acres.} \]
    • Thus, the Area = 0.00124 acres.

  3. Finding the Area of Field C using the Pythagorean Theorem:

    • The coordinates for Field C are (4, 4) and (7, 3).
    • The two points form a right triangle with the horizontal distance (x) and vertical distance (y) as legs of the triangle:
      • Horizontal distance \( |7 - 4| = 3 \)
      • Vertical distance \( |4 - 3| = 1 \)
    • Using the Pythagorean Theorem to find the hypotenuse (c): \[ c^2 = a^2 + b^2 \Rightarrow c^2 = 3^2 + 1^2 \Rightarrow c^2 = 9 + 1 = 10 \Rightarrow c = \sqrt{10}. \]
    • The area for Field C is half the base times height:
      • Base is 3 and Height is 1, hence: \[ \text{Area} = \frac{1}{2} \times 3 \times 1 = 1.5 \text{ square units.} \]
    • Converting to acres: \[ \text{Area in acres} = \frac{1.5}{4840} \approx 0.00031 \text{ acres.} \]
    • Thus, Area = 0.00031 acres.

Summary:

  1. Area of Field A = 0.00217 acres.
  2. Area of Field B = 0.00124 acres.
  3. Area of Field C = 0.00031 acres.
Are you sure those are the correct answers?

Let's reevaluate the problem step by step using the coordinates provided. The coordinates can indeed result in specific areas, so accuracy is key. Here's how to approach this task systematically.

Systematic Area Calculation:

  1. Field A Coordinates: (0, 0), (3, 0), (7, 3), (4, 7)

    • This forms a polygon that can be broken down into triangles to find the area.
    • The best way to find the area of a complex polygon is by using the shoelace formula, or breaking it down into simpler sections.
    • However, let’s visualize and calculate the area assuming it's a quadrilateral or triangle.

    To simplify:

    • We can calculate the area as: \[ \text{Area} = \frac{1}{2} | (x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1) - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) | \] With points \( (0, 0), (3, 0), (7, 3), (4, 7) \):

      • Substitute into the formula: \[ = \frac{1}{2} |(00 + 33 + 77 + 40) - (03 + 07 + 34 + 70)| \] \[ = \frac{1}{2} |(0 + 9 + 49 + 0) - (0 + 0 + 12 + 0)| \] \[ = \frac{1}{2} |58 - 12| = \frac{1}{2} \times 46 = 23 \text{ square units.} \]

    • Convert to Acres (1 acre = 4840 sq yards): \[ \text{Area in acres} = \frac{23}{4840} \approx 0.00475 \text{ acres.} \]

  2. Field B Coordinates: (0, 0), (0, -3), (-4, 0), (3, 0)

    • This forms another polygon. Using the shoelace method again: Using points \( (0, 0), (0, -3), (-4, 0), (3, 0) \): \[ = \frac{1}{2} |(0*(-3) + 00 + (-4)0 + 30) - (00 + (-3)(-4) + 03 + 0*0)| \] \[ = \frac{1}{2} |0 - 12| = \frac{1}{2} * 12 = 6 \text{ square units.} \]

    • Convert to Acres: \[ \text{Area in acres} = \frac{6}{4840} \approx 0.00124 \text{ acres.} \]

  3. Field C: Coordinates (4, 4) and (7, 3)

    • The calculation needs to analyze how the points perform as vertices of a right-angled triangle.

    • Finding the sides can be conducted as follows:

      • Length of one side (x direction): \[ |7 - 4| = 3 \]
      • Length of the other side (y direction): \[ |4 - 3| = 1 \]

    • Area: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 1 = 1.5 \text{ square units.} \]

    • Convert to Acres: \[ \text{Area in acres} = \frac{1.5}{4840} \approx 0.00031 \text{ acres.} \]

Final Summary of Areas:

  1. Area of Field A = 0.00475 acres.
  2. Area of Field B = 0.00124 acres.
  3. Area of Field C = 0.00031 acres.

Let me know if you need further calculations or clarifications!