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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 $B = 92, 12, C = − 1 2, 52$ .

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True

False

answer is A.

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Detailed Solution

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 = 92, 12, C = − 1 2, 52

.
Using the distance formula,

D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

We know, according to Pythagorean theorem if

Δ A B C

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,

A B = B C = C D = D A

Now,

A B = B C

For AB, substituting

x_{2} = x, x_{1} = 3, y_{2} = y, and y_{1} = 4

, in the formula,
For BC, substituting

x_{2} = 1, x_{1} = x, y_{2} = y, and y_{1} = (- 1)

, in the formula,

\Rightarrow AB = BC

\Rightarrow \sqrt{{(x - 3)}^{2} + {(y - 4)}^{2}} = \sqrt{{(1 - x)}^{2} + {(y - (- 1))}^{2}}

\Rightarrow x^{2} - 6 x + 9 + y^{2} - 8 y + 16 = 1 - 2 x + x^{2} + y^{2} + 2 y + 1

\Rightarrow 4 x + 10 y = 23

\Rightarrow x = \frac{23 - 10 y}{4}

Now put x = 23 − 10 y 4

Hence Point B = 92, 12 and point C = − 1 2, 52

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

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