As shown in the figure, the two diagonals of diamond ABCD are 6 and 8 long respectively. Point P is a moving point on diagonal AC, and points m and N are the midpoint of edge AB and BC respectively. Then the minimum value of PM + PN is______ ．

As shown in the figure, the two diagonals of diamond ABCD are 6 and 8 long respectively. Point P is a moving point on diagonal AC, and points m and N are the midpoint of edge AB and BC respectively. Then the minimum value of PM + PN is______ ．

As shown in the figure: make me ⊥ AC intersect ad to e, connect en, then en is the minimum value of PM + PN, ∵ m and N are the midpoint of AB and BC respectively, ∵ BN = BM = am, ∵ me ⊥ AC intersects ad to e, ∵ AE = am, ∵ AE = BN, AE ∥ BN, ∵ quadrilateral abne is parallelogram, ∵ en = AB, en ∥ AB, and from the meaning of the question, we can get AB = (6 ／ 2) 2 + (8 ／ 2) 2 = 5, ∥ en = AB = 5, ∥ PM + PN minimum value is 5