已知:如图,扇形OAB和扇形OA′B′的圆心角相同,设AA′=BB′=d.AB=l1,A′B′=l2.求证:图中阴影部分的面积S=12(l1+l2)d.-数学

题目简介

已知:如图,扇形OAB和扇形OA′B′的圆心角相同,设AA′=BB′=d.AB=l1,A′B′=l2.求证:图中阴影部分的面积S=12(l1+l2)d.-数学

题目详情

已知:如图,扇形OAB和扇形OA′B′的圆心角相同,设AA′=BB′=d.
AB
=l1
A′B′
=l2
求证:图中阴影部分的面积S=
1
2
(l1+l2)d

题型:解答题难度:中档来源:不详

答案

证明:设∠AOB=n°,OA′=OB′=r,
AB
=l1,
A′B′
=l2.
∴l1=
nπ(r+d)
180
,l2=class="stub"nπr
180

l1
l2
=class="stub"r+d
r

∴r=
l2d
l1-l2
①,
∵S阴影=S扇形OAB-S扇形OA′B′=class="stub"1
2
l1(r+d)-class="stub"1
2
l2r=class="stub"1
2
(l1r+l1d-l2r)
=class="stub"1
2
[(l1-l2)r+l1d]
=class="stub"1
2
[(l1-l2)×
l2d
l1-l2
+l1d]
=class="stub"1
2
(l2d+l1d)
=class="stub"1
2
(l1+l2)d.

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