sábado, 18 de junho de 2022

Geometria - Áreas e perímetros


Existem apenas cinco triângulos com lados inteiros em que a área é numericamente igual ao perímetro: (5, 12, 13); (6, 8, 10); (6, 25, 29); (7, 15, 20); (9, 10, 17). Apenas os dois primeiros são triângulos retângulos.

Os únicos retângulos com lados inteiros em que a área é numericamente igual ao perímetro são o quadrado 4 × 4 e o retângulo 3 × 6.

****

(There are only five integer-sided triangles in which the area numerically equals the perimeter: (5, 12, 13); (6, 8, 10); (6, 25, 29); (7, 15, 20); (9, 10, 17). Only the first two are right triangles.

The only rectangles with integer sides in which the area numerically equals the perimeter are the 4 × 4 square and the 3 × 6 rectangle.)

PCFilho

4 comentários:

  1. The rectangular case is easier to prove, so we’ll start with that.

    Let x and y be the lengths of the base and height, respectively, of a rectangle. Thus we have the following formulae for the perimeter and area:
    P = 2x + 2y, A = xy
    Since both expressions are symmetric, we can assume that x < y without loss of generality.
    Equating the right sides and solving for y, we get this:
    xy = 2x + 2y
    (x – 2)(y) = 2x
    y = 2x/(x – 2) = 2 + 4/(x – 2)
    From this equation, we see that y is an integer only when x – 2 divides 4; that is, when x is 3, 4, or 6. Plugging these values for x into the equation, we get the following results:
    x = 3 → y = 2 + 4/1 = 6
    x = 4 → y = 2 + 4/2 = 4
    x = 6 → y = 2 + 4/3 = 3 (we dismiss this result because it contradicts x < y)
    Therefore the only two rectangles with integer side lengths and whose perimeter and area are numerically equal are the 3 x 6 rectangle and the 4 x 4 rectangle.
    QED

    As for the triangular case, this proof is a little bit more difficult.
    Let the sides of the triangle be a, b, and c. The perimeter and area formulae are:
    P = a + b + c, A = (1/4)√[(a + b + c)(a + b – c)(a + c – b)(b + c – a)]
    Again, both expressions are symmetric, so we can assume a < b < c without loss of generality.
    Let’s simplify the expressions by defining a, b, and c in terms of some new variables thusly:
    a = x + y, b = x + z, c = y + z
    Thus:
    a + b + c = 2(x + y + z)
    a + b – c = 2x
    a + c – b = 2y
    b + c – a = 2z
    Then the perimeter and area formulae become:
    P = 2(x + y + z), A = √[(x + y + z)(xyz)]
    Equating the right sides and solving for z:
    2(x + y + z) = √[(x + y + z)(xyz)]
    2√(x + y + z) = √(xyz)
    4(x + y + z) = xyz
    4(x + y) = (xy – 4)z
    z = 4(x + y)/(xy – 4) = (x + y)[4/(xy – 4)]
    Thus z will be guaranteed to be an integer if xy – 4 divides 4. So we’re looking for values of x and y such that their product is 5, 6, or 8. There are five such combinations of (x,y) such that x < y: (1,5); (1,6); (1,8); (2,3); and (2,4). This leads to the following results:
    (x,y) = (1,5) → z = 6(4/1) = 24 → a = 6, b = 25, c = 29
    (x,y) = (1,6) → z = 7(4/2) = 14 → a = 7, b = 15, c = 20
    (x,y) = (1,8) → z = 9(4/4) = 9 → a = 9, b = 10, c = 17
    (x,y) = (2,3) → z = 5(4/2) = 10 → a = 5, b = 12, c = 13
    (x,y) = (2,4) → z = 6(4/4) = 6 → a = 6, b = 8, c = 10
    Therefore, these five triangles are the only ones with integer side lengths and whose perimeter and area are numerically equal.
    QED

    ResponderExcluir
  2. Notice how both proofs involve the factors of 4 somehow.

    ResponderExcluir
    Respostas
    1. Thank you, Jake. Amazing demonstration. :)

      Excluir
    2. You're welcome, although I *did* make a little mistake: Where the first proof has "2 + 4/3", it should say "2 + 4/4".

      Excluir

Regras para postar comentários:

I. Os comentários devem se ater ao assunto do post, preferencialmente. Pense duas vezes antes de publicar um comentário fora do contexto.

II. Os comentários devem ser relevantes, isto é, devem acrescentar informação útil ao post ou ao debate em questão.

III. Os comentários devem ser sempre respeitosos. É terminantemente proibido debochar, ofender, insultar e/ou caluniar quaisquer pessoas e instituições.

IV. Os nomes dos clubes devem ser escritos sempre da maneira correta. Não serão tolerados apelidos pejorativos para as instituições, sejam quais forem.

V. Não é permitido pedir ou publicar números de telefone/Whatsapp, e-mails, redes sociais, etc.

VI. Respeitem a nossa bela Língua Portuguesa, e evitem escrever em CAIXA ALTA.

Os comentários que não respeitem as regras acima poderão ser excluídos ou não, a critério dos moderadores do blog.