Odpowiedź:
A = ( -3 , - 1 ) , B = ( 5 , - 3 ) , C = ( 2 , 4 )
xa = - 3 , xb = 5 , xc = 2 , ya = - 1 , yb = - 3 , yc = 4
Obliczamy prostą przechodzącą przez punkty A i B
(xb - xa)(y - ya) = (yb - ya)(x - xa)
(5 + 3)(y + 1)= (- 3 + 1)(x + 3)
8(y + 1)= - 2(x+ 3)
8y + 8 = - 2x - 6
8y = - 2x - 6 - 8
8y = - 2x - 14
y = (-2/8)x - 14/8
y = (-1/4)x - 1 6/8
y = (-1/4)x - 1 3/4
Postać ogólna prostej
Ax + By + C = 0
(1/4)x + y + 1 3/4 = 0
A = 1/4 , B = 1 , C = 1 3/4 = 7/4 ; xc = 2 , yc = 4
d - odległość punktu C od prostej = IAxc + Byc + CI/√(A²+ B²) =
= I1/4 * 2 + 1 * 4 + 7/4I/√[(1/4)² + (7/4)²] = I1/2 + 4 + 7/4I/√(1/16 + 49/16) =
= I2/4 + 4 + 7/4I/√(50/16) = (9/4 + 4I/√[(√25 *2)/4] =
= I4 + 2 1/4I : 5√2/4 = I6 1/4I : 5√2/4 = 6 1/4 : 5√2/4 = 25/4 * 4/5√2 =
= 5/√2 = 5√2/2 [j]
P - pole trójkąta = 1/2I(xb - xa)(yc - ya) - (yb - ya)(xc - xa)I =
= 1/2I(5 + 3)(4 + 1) - (- 3 + 1)(2 + 3)I = 1/2I8 * 5 - (- 2) * 5I = 1/2I40 + 10I =
= 1/2I50I = 1/2 * 50 = 25 [j²]