<Qua-2025-765>, hint

Let E be the midpoint of BC, then DE pass through N , then PN=CN <DNC=90

Hence DP=DC. Lets calculate  the length of AN

A(0,2) c(2,0), N(6/5, 2/5) then AN=AD=2. AQ pass through the midpoint F of DN

and <AFD=<MND=90, hence QF//PN Hence QN//CD, QN=1/2 (CD)