Math Problem Statement
A surveyor needs to determine the area of a land show in Fig below. The coordinates of the four vertices of the land are as follows: A(8,13), B(3,10) , C(4,4), D(16,5)
Solution
To determine the area of a land plot with vertices given by their coordinates, we can use the formula for the area of a polygon when the coordinates of its vertices are known. This formula is known as the Shoelace formula or Gauss's area formula. For a polygon with vertices , the area is given by:
For the given vertices , , , and , we will calculate the area as follows:
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List the coordinates cyclically:
(8, 13) & (3, 10) & (4, 4) & (16, 5) & (8, 13) \\ \end{array}$$ -
Apply the Shoelace formula:
So, the area of the land is square units.
Would you like further details or have any questions?
Here are five related questions you might find interesting:
- How would the area calculation change if the coordinates were given in a different order?
- What is the significance of the absolute value in the Shoelace formula?
- How can the Shoelace formula be derived?
- Can the Shoelace formula be used for polygons with more than four sides?
- What are some real-life applications of the Shoelace formula?
Tip: Always double-check your vertices order when applying the Shoelace formula to ensure accurate area calculation.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Formulas
Shoelace formula
Gauss's area formula
Theorems
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Suitable Grade Level
Grades 9-12
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