State whether the given statement is true or false.If A(3, 4) and D(1, -1) are the angular points on the opposing sides of a square, then the remaining angular points coordinates are

B=

9
2

,
1
2

,C=

−1
2

,
5
2

.

Question

State whether the given statement is true or false.If A(3, 4) and D(1, -1) are the angular points on the opposing sides of a square, then the remaining angular points coordinates are

B=

9
      2
     
     ,
      1
      2

,C=

−1
      2
     
     ,
      5
      2

.

Accepted Answer

We have been given,If A(3, 4) and D(1, -1) are the angular points on the opposing sides of a square, then the remaining angular points coordinates are

B=

9
      2
     
     ,
      1
      2

,C=

−1
      2
     
     ,
      5
      2

.Using the distance formula, D=x2-x12+y2-y12 We know, according to Pythagorean theorem if

ΔABC
  is a right angled triangle and B is the right angle then,

A
    B
    2
   
   +B
    C
    2
   
   =A
    C
    2

Currently, the angles of a square's diagonals are,

(3,4)
  and

(1,−1)
  Hence,

A=
    
     3,4
   ,C=
    
     1,−1
   
  Let the coordinates of B be,

x,y
   
   Since it is a square, Therefore,

AB=BC=CD=DA
  Now,

AB=BC
  For AB, substituting x2=x, x1=3, y2=y, and y1=4, in the formula,For BC, substituting x2=1, x1=x, y2=y, and y1=(-1), in the formula,⇒AB=BC⇒(x-3)2+(y-4)2=(1-x)2+(y-(-1))2⇒x2-6x+9+y2-8y+16=1-2x+x2+y2+2y+1⇒4x+10y=23⇒x=23-10y4

Now put x=
    
     23−10y
    4

Hence Point B=

9
      2
     
     ,
      1
      2

and point C=

−1
      2
     
     ,
      5
      2

Therefore, our statement is true.Hence, option 1 is correct.

State whether the given statement is true or false.

If A(3, 4) and D(1, -1) are the angular points on the opposing sides of a square, then the remaining angular points coordinates are $B = 92, 12, C = − 1 2, 52$ .

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State whether the given statement is true or false.If A(3, 4) and D(1, -1) are the angular points on the opposing sides of a square, then the remaining angular points coordinates are B= 9 2 , 1 2 ,C= −1 2 , 5 2 .

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State whether the given statement is true or false.

If A(3, 4) and D(1, -1) are the angular points on the opposing sides of a square, then the remaining angular points coordinates are $B = 92, 12, C = − 1 2, 52$ .