As shown in the figure, in the diamond ABCD, ∠ a = 60 °, points P and Q are on edges AB and BC respectively, and AP = BQ Given ad = 3, AP = 2, find the length of PQ

As shown in the figure, in the diamond ABCD, ∠ a = 60 °, points P and Q are on edges AB and BC respectively, and AP = BQ Given ad = 3, AP = 2, find the length of PQ

(1) ∵ in diamond ABCD, ∠ a = 60 °
The angle of ABC is 120 ° and BD bisects ABC and △ abd is equilateral triangle
∴∠DBC =60°,AD=BD
∴∠DBC =∠A
∵AP=BQ
∴△BDQ≌△ADP
(2) Make QE ⊥ AB intersect AB extension line and point e through point Q (as shown in the figure)
∵ quadrilateral ABCD is diamond
∴AB=AD=3
∵AP=2
∴BP=1,BQ=AP=2
∠CBE=180°－120°=60°
Be = 1, QE = root 3
Ψ PE = 2, PQ = under root (2 & # 178; + (root 3) &# 178; = root 7)
Ψ cos ∠ bpq = PE / PQ = 2 / radical 7 = 2 · radical 7 / 7
Please accept if you are satisfied