The length of each side of a square ABCD is 12 cm, P is on BC, BP=5 is 5 cm, EF is perpendicular to AP at Q, and intersects AB and CD at E and F respectively. Find the length of EF.

Solution: the intersection of e point is EG//BC, and the intersection of CD point is g point.

∵ABCD is a square.

∴∠B=∠C=90，AB//CD，AB=BC

△ in △ABP, ∠ BAP+∠ APB = 90, AB 2+BP 2 = AP 2.

∫AB = 12cm，BP=5cm

∴AP= 13cm

∵EF is perpendicular to AP in Q.

∴∠AQE=90

∴, ∠ EAQ+∠ AEQ = 90 in △AEQ.

∠∠BAP =∠EAQ

∴∠APB=∠AEQ

∫AB//CD

∴∠EFG=∠AEQ

∴∠APB=∠EFG

∫EG//BC，∠B=∠C=90

∴∠C=∠FGE=∠B=90

∴BCGE is a rectangle.

∴BC=EG

AB = BC

∴AB=EG

∠∠APB =∠EFG，∠B=∠FGE，AB=EG

∴△APB≌△EFG(AAS)

∴AP=EF= 13cm