Area and perimeter in the coordinate plane: Set 2

Learning Objectives Calculate the perimeter of complex polygons using the Distance Formula. Determine the area of any simple polygon using the Shoelace Formula. Find the area of composite figures by decomposing them into simpler shapes like triangles and rectangles. Apply the 'boxing-in' method to find the area of non-standard polygons. Verify properties of geometric figures using coordinate geometry formulas before calculating area and perimeter. Solve multi-step problems involving area and perimeter on the coordinate plane. Ever wondered how a GPS calculates the area of a park or a property lot just from its corner coordinates? 🗺️ Let's learn the powerful geometry behind it! This tutorial moves beyond simple squares and focuses on complex and composite shap...

TermDefinitionExample Coordinate PolygonA polygon whose vertices are defined by ordered pairs (x, y) in the Cartesian coordinate plane.A triangle with vertices at A(1, 2), B(5, 6), and C(3, -1). Distance FormulaA formula derived from the Pythagorean theorem used to find the straight-line distance between two points in the coordinate plane.The distance between (2, 3) and (5, 7) is sqrt((5-2)^2 + (7-3)^2) = sqrt(3^2 + 4^2) = sqrt(25) = 5 units. Shoelace FormulaA method to calculate the area of any simple polygon given the coordinates of its vertices in order.For a triangle with vertices (x1, y1), (x2, y2), (x3, y3), the area is 0.5 * |(x1y2 + x2y3 + x3y1) - (y1x2 + y2x3 + y3x1)|. Decomposition MethodThe process of finding the area of a complex polygon by breaking it down into simpler, non-o...

The Distance Formula d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Use this formula to find the length of any segment in the coordinate plane, which is essential for calculating the perimeter of a polygon. (x₁, y₁) and (x₂, y₂) are the coordinates of the two endpoints. The Shoelace Formula Area = \frac{1}{2} |(x_1y_2 + x_2y_3 + ... + x_ny_1) - (y_1x_2 + y_2x_3 + ... + y_nx_1)| A powerful formula for finding the area of any simple polygon. List the n vertices (x₁, y₁), ..., (xₙ, yₙ) in counter-clockwise order and repeat the first vertex at the end. Sum the products of the downward diagonals and subtract the sum of the products of the upward diagonals. Area of a Triangle (Coordinate Method) Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| This is a simp...