Question

Determine x para que se tenha ab=cd, sendo A (x, 1), B (4, x+3), C (x, x+2) e D (2x, x+6):

Answer 1

Sabemos que:

A = (x, 1)
B = (4, x + 3)
C = (x, x + 2)
D = (2x, x + 6)

d(AB) = d(CD)

Logo:

√[(x + 3 - 1)² + (4 - x)²] = √[(x + 6 - (x + 2))² + (2x - x)²]

Podemos cortar as raízes, logo:

(x + 2)² + (4 - x)² = (x + 6 - x - 2)² + x²
x² + 4x + 4 + 16 - 8x + x² = 16 + x²
2x² - 4x + 20 = 16 + x²
2x² - x² - 4x + 20 - 16 = 0
x² - 4x + 4 = 0

∆ = (-4)² - 4(1)(4)
∆ = 16 - 16 = 0
√∆ = 0

x = (-b ± √∆)/2a
x = (-(-4) ± 0)/2
x = 4/2
x = 2

Resposta: x = 2.