Perimeter and Area with Coordinates
Aligned to G-GPE.B.7 — Common Core State Standards for Mathematics.
What this lesson teaches
With vertices given as coordinates, use the distance formula to find each side length, then add them for perimeter. For area of a triangle or rectangle, find a base and a perpendicular height from the coordinates, or use a known formula once dimensions are computed.
Worked examples
Example 1
Find the perimeter of the triangle with vertices A(0,0), B(3,0), C(3,4). Side AB = 3 (horizontal). Side BC = 4 (vertical). Side CA = sqrt((3-0)^2 + (4-0)^2) = sqrt(9 + 16) = sqrt(25) = 5. Perimeter = 3 + 4 + 5 = 12 units. The right angle at B also gives area = (1/2)(3)(4) = 6 square units.
Example 2
Find the area of the quadrilateral A(0,0), B(5,0), C(5,3), D(0,6). It is a trapezoid with the two vertical sides parallel: BC = 3 and AD = 6, with the horizontal distance between them 5. Area = (1/2)(b1 + b2)(h) = (1/2)(3 + 6)(5) = (1/2)(9)(5) = 22.5 square units.
Example 3
Common mistake: using the slanted side as the height. In triangle A(0,0), B(6,0), C(2,4), a student may take CA = sqrt(4 + 16) = 4.47 as the height against base AB. The height must be PERPENDICULAR to the base: here C sits 4 units above the x-axis, so h = 4 and the area is (1/2)(6)(4) = 12, not (1/2)(6)(4.47) = 13.4. Read the height off the perpendicular distance, never off a side length.
Practice questions
- Find the length of the segment from (1, 2) to (4, 6).
- Find the perimeter of the rectangle with vertices (0,0), (5,0), (5,2), (0,2).
- Find the area of the triangle with vertices (0,0), (6,0), (0,8).
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Perímetro y área con coordenadas
Con los vértices dados como coordenadas, usa la fórmula de distancia para hallar cada lado, luego súmalos para el perímetro. Para el área de un triángulo o rectángulo, halla una base y una altura perpendicular a partir de las coordenadas, o usa una fórmula conocida una vez calculadas las dimensiones.
Ejemplo 1: Halla el perímetro del triángulo con vértices A(0,0), B(3,0), C(3,4). Lado AB = 3 (horizontal). Lado BC = 4 (vertical). Lado CA = raíz((3-0)^2 + (4-0)^2) = raíz(9 + 16) = raíz(25) = 5. Perímetro = 3 + 4 + 5 = 12 unidades. El ángulo recto en B también da área = (1/2)(3)(4) = 6 unidades cuadradas.
Ejemplo 2: Halla el área del cuadrilátero A(0,0), B(5,0), C(5,3), D(0,6). Es un trapecio con los dos lados verticales paralelos: BC = 3 y AD = 6, con distancia horizontal 5 entre ellos. Área = (1/2)(b1 + b2)(h) = (1/2)(3 + 6)(5) = (1/2)(9)(5) = 22.5 unidades cuadradas.
Ejemplo 3: Error común: usar el lado inclinado como altura. En el triángulo A(0,0), B(6,0), C(2,4), alguien puede tomar CA = sqrt(4 + 16) = 4.47 como altura sobre la base AB. La altura debe ser PERPENDICULAR a la base: aquí C está 4 unidades por encima del eje x, así que h = 4 y el área es (1/2)(6)(4) = 12, no (1/2)(6)(4.47) = 13.4. Lee la altura como distancia perpendicular, nunca como la longitud de un lado.
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