△ABC的外接圆圆心为O，且3OA+4OB+5OC＝0，则∠C等于．

Question

△ABC的外接圆圆心为O，且3

OA

+4

OB

+5

OC

＝

0

，则∠C等于______．

Answer 1

∵△ABC的外接圆圆心为O，且3

OA

+4

OB

+5

OC

＝

0

，
∴|

OA

| ＝|

OB

| ＝|

OC

|，

OC

＝−(1)/(5)(3 

OA

+4

OB

)，
∴

OC

•

OC

＝|

OC

|2
=(1)/(25)(3

OA

+4

OB

)2
=(9)/(25)|

OA

|2+(24)/(25)

OA

•

OB

+(16)/(25)|

OB

|2
=|

OC

|2+(24)/(25)

OA

•

OB

，
∴(24)/(25)

OA

•

OB

=0，∴∠AOB=90°．外接圆中，OA=OB，
∴O为AC中点，
∵∠B为90°，
∴∠C=45°．
故答案为：45°．