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△ABC的外接圆圆心为O,且3OA+4OB+5OC=0,则∠C等于.

题目详情
△ABC的外接圆圆心为O,且3
OA
+4
OB
+5
OC
0
,则∠C等于______.
▼优质解答
答案和解析
∵△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°.
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