Math Problem Statement
Solution
Let's solve the problem step by step. The given shape is a polygon with vertices labeled , and the coordinates are:
Step 1: Finding the Area
We will use the Shoelace Theorem (or surveyor's formula) for finding the area of a polygon with known coordinates. The formula is:
Substituting the points :
Calculating step by step:
- First part:
- Second part:
Now subtract and take the absolute value:
Thus, the area is 26 square units.
Step 2: Finding the Perimeter
The perimeter is the sum of the lengths of the sides. To find the length of each side, we will use the distance formula:
- :
- :
- :
- :
- :
Now, sum up all the lengths:
This simplifies to:
So, the perimeter is units.
Final Answer:
- Area = 26 square units
- Perimeter = units
Let me know if you need any clarifications or further details!
Here are 5 questions to expand on this topic:
- What is the Shoelace Theorem and how does it work for polygons?
- How would you find the perimeter of a polygon if it had more sides?
- Can the Shoelace Theorem be used for any type of polygon, and why?
- What is the difference between exact values like and decimal approximations?
- How would you approach calculating the area of a shape that is not on a grid?
Tip: When calculating the perimeter of polygons, always ensure to use the distance formula for each side, especially if the polygon is irregular or has diagonals.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Polygons
Formulas
Shoelace Theorem for area: Area = 1/2 * |x1y2 + x2y3 + ... + xny1 - (y1x2 + y2x3 + ... + ynx1)|
Distance formula for perimeter: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Theorems
Shoelace Theorem
Suitable Grade Level
Grades 9-12
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