Question

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Two opposite angular points of a square ABCD are A(-1,2) and C(3,-2) .Find the coordinates of the remaining angular points of the square ?

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Answer 1

Answer:

Let ABCD be a square and let A(3,4) and C(1,-1) be the given angular points. Let B(x,y) be the unknown vertex.

Then, AB=BC

⇒AB 2

=BC 2

⇒(x−3) 2+(y−4) 2

=(x−1) 2+(y+1) 2

⇒4x+10y−23=0

⇒x= 4

23−10y

....(i)

In right-angled triangle ABC, we have

AB 2 +BC 2

=AC 2

⇒(x−3) 2 +(y−4) 2 +(x−1) 2 +(y+1) 2

=(3−1) 2 +(4+1) 2

⇒x 2 +y 2 −4x−3y−1=0 ...(ii)

Substituting the value of x from (i) into (ii), we get

( 423−10y ) 2 +y 2 −(23−10y)−3y−1=0

⇒4y 2 −12y+5=0

⇒(2y−1)(2y−5)=0

⇒y= 21 or, 25

Puttingy= 21and y= 25

respectivelyin(i),weget

x= 29 and X= 2−1

respectively.

Hence, the required vertices of the square are (9/2,1/2) and (-1/2,5/2).

Answer 2

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Two opposite angular points of a square ABCD are A(-1,2) and C(3,-2) .Find the coordinates of the remaining angular points of the square ?

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The two opposite vertices of a square are (-1 , 2) and (3 , 2). Find the co-ordinates of the other two vertices.

Let me answer a more general question.

If two opposite vertices of a square are

A(2a,2b)

and

C(2c,2d),

then the coordinates of the other two vertices, in counterclockwise order, are

B(a−b+c+d,a+b−c+d)(1)

and

D(a+b+c−d,−a+b+c+d)(2)

To see that ABCD is a square, calculate the vectors

AB→=⟨−a−b+c+d,a−b−c+d⟩

BC→=⟨−a+b+c−d,−a−b+c+d⟩

CD→=⟨a+b−c−d,−a+b+c−d⟩

DA→=⟨a−b−c+d,a+b−c−d⟩

and then note that the dot products AB→⋅BC→, BC→⋅CD→, CD→⋅DA→, and DA→⋅AB→ are all zero, and the squared distances (vector norms) AB, BC, CD, and DA are all equal to 2(a−c)2+2(b−d)2. To see that the vertices are named in counterclockwise order, note that the cross product AB→×BC→ is equal to (2(a−c)2+2(b−d)2)k^. From the fact that the k^ component is positive, the right-hand-rule of cross products tells us that A, B, and C are named in counterclockwise order.

Now, for this problem, we start with

A(−1,2)

and

C(3,2),

and then using the formulas (1) and (2), above, we calculate

B(1,0)

and

D(1,4)

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