Polygons and Algebraic Relationships

Lesson 15

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Objective

Calculate and justify the area and perimeter of polygons on the coordinate plane given a system of inequalities. 

Common Core Standards

Core Standards

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  • G.GPE.B.4 — Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

  • G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • A.REI.D.12 — Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Foundational Standards

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  • 6.G.A.3

  • 8.G.B.8

Criteria for Success

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  1. Create a polygon by graphing a system of inequalities. 
  2. Prove a given system of inequalities creates a specific polygon. 
  3. Find the area and perimeter given a system of inequalities.

Tips for Teachers

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Anchor Problems

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Problem 1

$${\mathrm{Given}\left\{\begin{matrix}y \leq4 \\ y \geq0 \\ y \leq 2x \\ y + 8 \geq 2x \end{matrix}\right.}$$

 

  1. Graph the system of inequalities. 
  2. Prove that the polygon created is a parallelogram.

Guiding Questions

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Problem 2

$${\mathrm{Given}\left\{\begin{matrix}y \leq -3x+8 \\ y \leq x \\ 3y \geq x-6 \end{matrix}\right.}$$

 

  1. Show that the triangle created is a right triangle. 
  2. Find the area of the triangle.
  3. Find the perimeter of the triangle. Round your answer to the nearest tenth.

Guiding Questions

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Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

  • Include problems where students are given a system of inequalities and asked to find the area and perimeter of a polygon. 
  • Include problems where students are given a system of inequalities and must show that the polygon created is a parallelogram, rectangle, rhombus, and/or square. 

Target Task

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  1. Determine the polygon described by the given system of inequalities. Explain your reasoning.
  2. Find the perimeter. Round your answer to the nearest tenth. 
  3. Find the area in simplest radical form.