Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities

New York State Common Core Math Geometry, Module 4, Lesson 11

Worksheets for Geometry

Student Outcomes

• Students find the perimeter of a triangle or quadrilateral in the coordinate plane given a description by inequalities.
• Students find the area of a triangle or quadrilateral in the coordinate plane given a description by inequalities by employing Green’s theorem.

Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities

Classwork

Opening Exercise

Graph the following:
a. 𝑦 ≤ 7
b. 𝑥 > −3
c. 𝑦 < 1/2 𝑥 − 4
d. 𝑦 ≥ −2/3 𝑥 +5

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Example 1

A parallelogram with base of length 𝑏 and height ℎ can be situated in the coordinate plane, as shown. Verify that the shoelace formula gives the area of the parallelogram as 𝑏ℎ.

Example 2

A triangle with base 𝑏 and height ℎ can be situated in the coordinate plane, as shown. According to Green’s theorem, what is the area of the triangle?

Exercises

1. A quadrilateral region is defined by the system of inequalities below:
   𝑦 ≤ 𝑥 + 6
   𝑦 ≤ −2𝑥 + 12
   𝑦 ≥ 2𝑥 − 4
   𝑦 ≥ −𝑥 + 2
   a. Sketch the region.
   b. Determine the vertices of the quadrilateral.
   c. Find the perimeter of the quadrilateral region.
   d. Find the area of the quadrilateral region.
2. A quadrilateral region is defined by the system of inequalities below:
   𝑦 ≤ 𝑥 + 5
   𝑦 ≥ 𝑥 − 4
   𝑦 ≤ 4
   𝑦 ≥ −5/4 𝑥 − 4
   a. Sketch the region.
   b. Determine the vertices of the quadrilateral.
   c. Which quadrilateral is defined by these inequalities? How can you prove your conclusion?
   d. Find the perimeter of the quadrilateral region.
   e. Find the area of the quadrilateral region.

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