If two opposite angular points of a square be (-1,3) and (5,3), find the co-ordinates of the remaining angular points
Answers
Answer:
Let ABCD is a square.
Let the coordinate of vertices A and C are (3, 4) and (1, -1) respectively.
Let the coordinate of vertex B be (x, y)
Since, in square, length of all sides are equal
=> AB = BC
=> √{(x - 3)2 + (y - 4)2 } = √{(x - 1)2 + (y + 1)2 }
Squaring on both side, we get
=> (x - 3)2 + (y - 4)2 = (x - 1)2 + (y + 1)2
=> x2 - 6x + 9 + y2 - 8y + 16 = x2 - 2x + 1 + y2 + 2y + 1
=> x2 - 2x + 1 + y2 + 2y + 1 - x2 + 6x - 9 - y2 + 8y - 16 = 0
=> 4x + 10y - 23 = 0
=> 4x + 10y = 23
Now in right angle triangle ABC,
AB2 + BC2 = CA2
=> (x - 3)2 + (y - 4)2 + (x - 1)2 + (y + 1)2 = (3 - 1)2 + (4 + 1)2
=> x2 - 6x + 9 + y2 - 8y + 16 + x2 - 2x + 1 + y2 + 2y + 1 = 4 + 25
=> 2x2 + 2y2 - 8x - 6y - 2 = 0
=> x2 + y2 - 4x - 3y - 1 = 0
=> {(23 - 10y)/4}2 + y2 - 4{(23 - 10y)/4} - 3y - 1 = 0
=> 529 + 100y2 - 460y + 16y2 - 368 + 160y - 48y - 16 = 0
=> 116y2 - 348y + 145 = 0
=> 29(4y2 - 12y + 5) = 0
=> 4y2 - 12y + 5 = 0
=> 4y2 - 10y - 2y + 5 = 0
=> 2y(2y - 5) - 1(2y - 5) = 0
=> (2y - 5) * (2y - 1) = 0
=> y = 1/2, 5/2
When y = 1/2, we get
x = (23 - 10 * 1/2)/4 = (23 - 5)/4 = 18/4 = 9/2
When y = 5/2, we get
x = (23 - 10 * 5/2)/4 = (23 - 25)/4 = -2/4 = -1/2
Step-by-step explanation:
Let ABCD is a square.
Let the coordinate of vertices A and C are (3, 4) and (1, -1) respectively.
Let the coordinate of vertex B be (x, y)
Since, in square, length of all sides are equal
=> AB = BC
=> √{(x - 3)2 + (y - 4)2 } = √{(x - 1)2 + (y + 1)2 }
Squaring on both side, we get
=> (x - 3)2 + (y - 4)2 = (x - 1)2 + (y + 1)2
=> x2 - 6x + 9 + y2 - 8y + 16 = x2 - 2x + 1 + y2 + 2y + 1
=> x2 - 2x + 1 + y2 + 2y + 1 - x2 + 6x - 9 - y2 + 8y - 16 = 0
=> 4x + 10y - 23 = 0
=> 4x + 10y = 23
Now in right angle triangle ABC,
AB2 + BC2 = CA2
=> (x - 3)2 + (y - 4)2 + (x - 1)2 + (y + 1)2 = (3 - 1)2 + (4 + 1)2
=> x2 - 6x + 9 + y2 - 8y + 16 + x2 - 2x + 1 + y2 + 2y + 1 = 4 + 25
=> 2x2 + 2y2 - 8x - 6y - 2 = 0
=> x2 + y2 - 4x - 3y - 1 = 0
=> {(23 - 10y)/4}2 + y2 - 4{(23 - 10y)/4} - 3y - 1 = 0
=> 529 + 100y2 - 460y + 16y2 - 368 + 160y - 48y - 16 = 0
=> 116y2 - 348y + 145 = 0
=> 29(4y2 - 12y + 5) = 0
=> 4y2 - 12y + 5 = 0
=> 4y2 - 10y - 2y + 5 = 0
=> 2y(2y - 5) - 1(2y - 5) = 0
=> (2y - 5) * (2y - 1) = 0
=> y = 1/2, 5/2
When y = 1/2, we get
x = (23 - 10 * 1/2)/4 = (23 - 5)/4 = 18/4 = 9/2
When y = 5/2, we get
x = (23 - 10 * 5/2)/4 = (23 - 25)/4 = -2/4 = -1/2