Q1=Two opposite vertices of a square are (-1,2)and (3,2).Find the coordinates of other two vertices.
Q2=If two opposite vertices of a square of a square are (5,4)and (1,-6) ,find the coordinates of its remaining two vertices.
Dear Student!
Let ABCD be a square and A (–1, 2) and C (3, 2) be the given vertices. Let the coordinate of vertex B be (x, y).
AB = BC ( ABCD is a square)
⇒ AB2 = BC2
⇒ [x – (–1)] 2 + (y – 2)2 = (x – 3)2 + (y – 2)2 (Distance formula)
⇒ (x + 1)2 = (x – 3)2
⇒ x2 + 2x + 1 = x2 – 6x + 9
⇒ 2x + 6x = 9 – 1
⇒ 8x = 8
⇒ x = 1
In ΔABC, we have
AB2 + BC2 = AC2 (Pythagoras theorem)
⇒ 2AB2 = AC2 ( AB = BC)
⇒ 2[(x – (–1))2 + (y – 2)2] = (3 – (–1))2 + (2 – 2)2
⇒ 2[(x + 1)2 + (y – 2)2] = (4)2 + (0)2
⇒ 2[(1 + 1)2 + (y – 2)2] = 16 ( x = 1)
⇒ 2[ 4 + (y – 2)2] = 16
⇒ 8 + 2 (y – 2)2 = 16
⇒ 2 (y – 2)2 = 16 – 8 = 8
⇒ (y – 2)2 = 4
⇒ y – 2 = ± 2
⇒ y – 2 = 2 or y – 2 = –2
⇒ y = 4 or y = 0
Thus, the other two vertices of the square ABCD are (1, 4) and (1, 0).
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