Taishan cos photography

(1)∵ points m and n are AB, the midpoint of AC side, and point d is the midpoint of BC side.

∴DM, DN is the center line,

∫∠A = 90 °,

∴∠DMP=∠DNQ=∠MDN=90，

∫∠PDQ = 90，

∴∠PDM=90 -∠PDN，∠QDN=90-∠PDN，

∴∠PDM=∠QDN，

∴△pdm∽△qdn；

②∫AB = 6，AC=8，

∴DM=4,DN=3，

∫△PDM∽△QDN，

∴PMQN=DMDN=43，

∴QN=34PM，

If point P is on AM and point Q is on CN, PM=3-x, QN=34(3-x), y=CQ=CN-QN=4-34(3-x)=74+34x,

If point P is on MB and point Q is on NA, PM=x-3, QN=34(x-3), y=CQ=CN+QN=4+34(x-3)=74+34x,

The functional relationship is y=34x+74.

When point Q coincides with point A, that is, CQ=8, then 34x+74=8,

Solution: x=253,

The value range of ∴x is 0 ≤ x ≤ 253;

(3)∵ Point D is the midpoint of the hypotenuse BC of Rt△ABC,

∴DA=DC=5，

According to (2), point Q coincides with point A, and △CDQ is an isosceles triangle. X=253 at this time.

If CQ=CD=5 and QN=CQ-CN= 1, then PM=43QN=43.

x=AP=3+43= 133，

If CQ=DQ and the intersection Q is QE⊥DC, CE=52, cos∠C=CECQ,

In Rt△ABC, cos∠C=ACBC=