A = (3; -1) , B = (1; 2), C = (-3; 3), D = (-4; -2) pr. AC y = ax + b -1 = 3a + b 3 = -3a + b ---------------- 2 = 2b --> b = 1 3a = -1 -b = -1 -1 = -2 a = -2/3 y = (-2/3) x + 1 -------------------- pr BF prostopadła do pr AC a*a1 = -1 (-2/3)*a1 = -1 ---> a1 = 3/2 y = (3/2) x + b 2 = (3/2)*1 + b b = 2 - 3/2 = 1/2 y = (3/2) x + 1/2 ---------------------- Szukam punktu F y = (-2/3) x + 1 oraz y = (3/2) x + 1/2 (-2/3)x + 1 = (3/2) x + 1/2 / * 6 -4x + 6 = 9x + 3 9x + 4x = 6 - 3 13x = 3 x = 3/13 y = (-2/3)*(3/13) + 1 = -2/13 + 13/13 = 11/13 F = ( 3/13; 11/13) ----------------------- pr DE prostopadła do pr AC a*a2 = -1 (-2/3)*a2 = -1 ---> a2 = 3/2 y = (3/2) x + b1 -2 = (3/2)*(-4) + b1 -2 = -6 + b1 ---> b1 = -2 + 6 = 4 y = (3/2) x + 4 Szukam punktu E (3/2) x + 4 = (-2/3)x + 1 (3/2)x +(2/3)x = 1 -4 = -3 / 6 9x + 4x = -18 13 x = -18 x = -18/13 y = (3/2)*(-18/13) + 4 = -27/13 + 52/13 = 25/13 E = ( -18/13; 25/13 ) I AC I² = (-3-3)² + (3-(-1))² = (-6)² + 4² = 36 + 16 = 52 AC = √52 = 2√13 --------------------- I BF I² = (3/13 -1)² + (11/13 -2)² = (-10/13)² +(-15/13)² = = 100/169 + 225/169 = 325/169 = 25/13 BF = 5/√13 ------------------------------------------------------- I DE I² = (-18/13 +4)² + (25/13 +2)² = = (34/13)² + (51/13)² = 1156/169+ 2601/169 = 3757/169 = = 289/13 DE = 17/√13 ----------------------- P = (1/2)*[AC*BF + AC *DE] = (1/2)*AC*[BF + DE] P = (1/2)*2√13*[5/√13 + 17/√13] = √13*[22/√13] = 22 Odp. P = 22 j² ===================================================
