In rectangle $ABCD$,$AB = 9$ and $BC = 12$. Then,$BD = \ldots$

Find the point on the $X$-axis which is equidistant from $(5, 4)$ and $(-2, 3)$.

$A$ person begins his journey from point $A$. From $A$,he moves $800 \, m$ towards north to reach point $B$. From $B$,he moves $500 \, m$ towards east to reach point $C$. From $C$,he moves $400 \, m$ towards north to reach point $D$. Find the direct distance in $m$ from $A$ to $D$.

Find the point on the $Y$-axis which is equidistant from $(3, 1)$ and $(-2, 5)$.

If $Q(0, 1)$ is equidistant from $P(5, -3)$ and $R(x, 6)$,find the values of $x$. Also,find the distances $QR$ and $PR$.

Prove that $A (1, 7)$,$B (2, 4)$,and $C (5, 5)$ are the vertices of an isosceles right triangle.