Question

The ends of a diagonal of a square are $(2,-3)$ and $(-1,1).$ Another vertex of the square can be

• A. $(-\frac{3}{2},-\frac{5}{2})$
• B. $(\frac{5}{2},\frac{1}{2})$
• C. $(\frac{1}{2},\frac{5}{2})$
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A $(-\frac{3}{2},-\frac{5}{2})$
B $(\frac{5}{2},\frac{1}{2})$

Let $AC$ be one diagonal of the square given.
$AC=5$ midpoint of AC is $(\frac{1}{2},-1)$
slope of AC $=\frac{1-(-3)}{-1-2}=\frac{-4}{3}$
The other diagonal BD is perpendicular to AC thus its slope $=\frac{3}{4}=\tan \theta ⟹\cos \theta =\frac{4}{5},\sin \theta =\frac{3}{5}$
now B and D are $(\frac{1}{2}\pm \frac{5}{2}\times \frac{4}{5},-1\pm \frac{5}{2}\times \frac{3}{5})$
$(\frac{5}{2},\frac{1}{2})$ and$(\frac{-3}{2},\frac{-5}{2})$