CC Geometry H

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CC Geometry H Aim #8: How do we find perimeter and area of polygons in the Cartesian Plane? Do Now: Find the area of the triangle below using decomposition and then using the shoelace formula (also known as Green's theorem). 1) Given rectangle ABCD, a. Identify the vertices. b. Find the exact perimeter. c. Find the area using the area formula. d. List the vertices starting with A moving counterclockwise and apply the shoelace formula. Does the method work for quadrilaterals? 2) Calculate the area using the shoelace formula and then find the perimeter (nearest hundredth). 3) Find the perimeter (to the nearest hundredth) and the area of the quadrilateral with vertices A(-3,4), B(4,6), C(2,-3), and D(-4,-4). 4) A textbook has a picture of a triangle with vertices (3,6) and (5,2). Something happened in printing the book and the coordinates of the third vertex are listed as (-1, ). The answers in the back of the book give the area of the triangle as 6 square units. What is the y-coordinate of this missing vertex? 5) Find the area of the pentagon with vertices A(5,8), B(4,-3), C(-1,-2), D(-2,4), and E(2,6). 6) Show that the shoelace formula used on the trapezoid shown confirms the traditional formula for the area of a trapezoid: 1 (b1 + b2)h 2 D (x3,y) C (x2,y) A B (x ,0) (0,0) 1 7) Find the area and perimeter (exact and nearest hundredth) of the hexagon shown. Let's Sum it Up! To find the area of a polygon, you can use the shoelace formula, which is also referred to as Green's theorem. *Remember: pick a starting point and move in a counter-clockwise direction!* Name____________________ CC Geometry H Date __________ HW #8 1) Given triangle ABC with vertices, A(7, 4), B(1, 1), and C(9, 0). a. Find the exact perimeter. b. Find the area using the traditional area formula. c. Find the area using the shoelace formula. d. Explain why the shoelace formula might be more useful and efficient if you were just asked to find the area. 2) Given triangle ABC and quadrilateral DEFG, describe how you would find the area of each and why you would choose that method, and then find the areas. 3) Find the area and perimeter (nearest hundredth) of quadrilateral ABCD with vertices: A(6,5), B(2, -4), C(-5, 2), and D(-3, 6). 4) Find the area and perimeter (nearest hundredth) of pentagon ABCDE with vertices A(2,6), B(7,2), C(3,-4), D(-3,-2), and E(-2,4). Review: 1) Solve for x given BD = 4x + 2 and AE =6x + 8. Assume B is the midpoint of AC and D is the midpoint of CE. 2) In the diagram below of right triangle ΔACB, altitude CD is drawn to hypotenuse AB. If AB = 36 and AC = 12, what is the length of AD? C 12 A B D 36.

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Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities 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. Lesson Notes In previous lessons, students found the area of polygons in the plane using the “shoelace” method. In this lesson, we give a name to this method—Green’s theorem. Students will draw polygons described by a system of inequalities, find the perimeter of the polygon, and use Green’s theorem to find the area. Classwork Opening Exercises (5 minutes) The opening exercises are designed to review key concepts of graphing inequalities. The teacher should assign them independently and circulate to assess understanding. Opening Exercises Graph the following: a. b. Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities 135 Date: 8/28/14 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY d. c. Example 1 (10 minutes) Example 1 A parallelogram with base of length and height can be situated in the coordinate plane as shown. Verify that the shoelace formula gives the area of the parallelogram as . What is the area of a parallelogram? Base height . The distance from the -axis to the top left vertex is some number .
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