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If two opposite vertices of a square are (5,4) and (1, -6), then find the coordinates of its remaining two vertices.

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(8, − 3) and (− 2, 1)

(8, 3) and (− 2, 1)

(8, − 3) and (2, 1)

(− 8, − 3) and (− 2, 1)

answer is A.

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Detailed Solution

It is given that,
Opposite vertices of a square are (5,4) and (1, -6).
Question Image

Let,

A = (5, 4) and C = (1, − 6)

.
Since ABCD is a square,

∴ A B = B C = C D = D A

.
Suppose, the coordinate of B is (x, y).
We know that, the distance between any two points in a Cartesian plane

M (a, b)

and

N (c, d)

is given by

M N = (c − a) 2 + (d − b) 2

.
Then,

A C = (1 − 5) 2 + (− 4 − 6) 2 = (4) 2 + (− 10) 2 = 16 + 100 = 116

Since,

A B = B C = C D = D A = A C 2 ∴ A B = B C = C D = D A = 1162 ⇒ A B = B C = C D = D A = 58

Now,

A B = B C

⇒ (x − 5) 2 + (y − 4) 2 = (1 − x) 2 + (y − (− 6)) 2

⇒ x 2 − 10 x + 25 + y 2 − 8 y + 16 = 1 − 2 x + x 2 + y 2 + 12 y + 36

⇒ 8 x + 20 y = 4

⇒ y = 1 − 2 x 5

… (1)
In

∆ ABC

∠ B = 90 °

Then by using Pythagoras Theorem we get,
Question Image

Therefore,

B 8, − 3 and D − 2, 1 .

Hence, option (1) is correct.

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