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If ABCD is a square where A(2,3), B(6,7), C(10,1), then D=?:

(6,3)
(-6,-3)
(6,-3)
(6,5)

Explanation

We are given three vertices of a square ABCD:
$A (2, 3)$
$B (6, 7)$
$C (10, 1)$
We need to find $D (x, y)$ .
Step 1: Understanding the Square Properties
A square has equal sides and perpendicular diagonals. The diagonals also bisect each other at their midpoint.
Step 2: Find the Midpoint of Diagonal AC
The midpoint formula for two points ( $1$ $,$ $y$ $1$ $)$ and ( $2$ $,$ $y$ $2$ $)$ is:
$M = (x _{1} + x _{2/2}, y _{1} + y _{2/2})$ For diagonal AC:
$M = (2 + 10, 3 + 1/2)$ $M = (12/2, 4) = (6, 2)$ Step 3: Use Midpoint to Find D
Since diagonals bisect each other, midpoint M is also the midpoint of diagonal BD.
Using the midpoint formula for BD:
$M = (B _{x} + D _{x} \frac{B _{y} + D _{y/2}}{2,})$ Substituting B $6$ $,$ $7$ $)$ and M $6$ $,$ $2$ $)$

$(6, 2) = (6 + D _{x}, 7 + D _{y})$

Step 4: Solve for D

Equating the coordinates:

For x-coordinate:
$6 + D _{x/2} = 6$ $6 + D_{x} = 12$ $D_{x} = 6$
For y-coordinate:
$7 + D _{y} = 2$ $7 + D_{y} = 4$ $D_y = -3$

$D = (6, - 3)$

Correct option: (6, -3)

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