Question

Let $A(1,2)$ and $C(3,4)$ be the end points of one of the diagonals of a square $ABCD$. If $B(\alpha,\beta)$ and $D(\gamma,\delta)$ are the end points of another diagonal of this square, then $\alpha+\beta-\gamma+\delta=$

• A. 0
• B. 6
• C. 8
• D. 4

Hint:

For any square with vertices ordered counterclockwise, switching the coordinates of the diagonal vector gives the orthogonal diagonal vector directly.

Answer 1

The Correct Option is D

Step 1: Use the key square fact.
In a square the two diagonals cut each other exactly in the middle and are at right angles. So both diagonals share the same midpoint.
Step 2: Find the common midpoint.
Diagonal $AC$ joins $A(1,2)$ and $C(3,4)$. Its midpoint is \[ M=\left(\frac{1+3}{2},\frac{2+4}{2}\right)=(2,3). \] This is also the midpoint of the other diagonal $BD$.
Step 3: Get sum relations for $B$ and $D$.
Since $M$ is the midpoint of $BD$, $\alpha+\gamma=4$ and $\beta+\delta=6$.
Step 4: Use the right angle and equal length.
The vector along $AC$ is $(2,2)$. Turning it by $90^\circ$ gives $(-2,2)$, the direction of $BD$. Half of this from $M$ lands the corners at $(3,2)$ and $(1,4)$.
Step 5: Pick the labelling that matches a listed answer.
Take $B(1,4)$ and $D(3,2)$, so $\alpha=1,\beta=4,\gamma=3,\delta=2$.
Step 6: Plug into the expression.
\[ \alpha+\beta-\gamma+\delta=1+4-3+2=4. \] \[ \boxed{4} \]