△ABC内接于以O为圆心的圆,且3OA+4OB-5OC=0.则∠C=______°,cosA=______.-数学

题目简介

△ABC内接于以O为圆心的圆,且3OA+4OB-5OC=0.则∠C=______°,cosA=______.-数学

题目详情

△ABC内接于以O为圆心的圆,且3
OA
+4
OB
-5
OC
=
0
.则∠C=______°,cosA=______.
题型:填空题难度:中档来源:不详

答案

3
OA
+4
OB
-5
OC
=
0

3
OA
+4
OB
=5
OC

9
OA
2
+24
OA
OB
+16
OB
2
=25
OC
2

∵A,B,C在圆上
设OA=OB=OC=1
OA
OB
=0

根据 3
OA
+4
OB
=-5
OC
得出A,B,C三点在圆心的同一侧
∴根据圆周角定理知∠C=180°-class="stub"1
2
×
90°=135°
同理求出
OB
BC
=class="stub"4
5

cos∠BOC=
class="stub"4
5
1×1
=class="stub"4
5

∵∠A是∠BOC的一半
cosA=
3
10
10

故答案为:135°;
3
10
10

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