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Answer» We know, properties of square : (1) all sides are equal (2) angle between two sides = 90°
use this concept here, Let one unknown point of square is C(P, Q) given, A( 5, 4) , D(-1, 6)
AC = DC AC² = DC² (P-5)² + (Q -4)² = (P+1)² + (Q -6)² -10P +25 -8Q + 16 = 2P +1 -12Q + 36 -12P + 4Q + 4 = 0 -3P + Q + 1 = 0 ________(1)
AC² + DC² = AD² (P+1)² + (Q -6)² + (P-5)² + (Q -4)² = (6)² + (2)² 2P² + 2Q² -8P -20Q + 78 = 40 P² + Q² -4P - 10Q + 19 = 0-----------(2) from (1) and (2)
P² + (3P-1)² -4P -10(3P-1) + 19 = 0 10P² +1 - 6P -4P -30P +10 + 19 = 0 10P² - 40P + 30 = 0 P² -4P + 3 = 0 P = 1 and 3 Q = 3P -1 = 2 and 8 hence, two unknown points of square are ( 1 , 2) and (3, 8)
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