Circles A. B, and C cach have raditus 1. Circles

Rupert Albert 2022-03-03 Answered
Circles A. B, and C cach have raditus 1. Circles A and B share one point of tangency. Circle C has a point of tangency with the midpoint of AB. What is the area inside circle C but outside circle A and circle B?

a) 3π2
b) π2
c) 2
d) 3π4
e) 1+π2
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Expert Answer

Pooja Copeland
Answered 2022-03-04 Author has 8 answers

Stpe 1
Given, Circles A, B, and C each have radius 1. Circles A and B share one point of tangency.
Circle C has a point of tangency with the midpoint of AB.
To find the are inside circle C but outside A and circle B, that is, the shaded region of the circle C.
Now, Area of shaded region=Area of Circle CArea shared between Circles A, B and C.
Let S be the mid point of AB and T be the point of intersection of Circles A and C.
Join the points by joining the points ST
Step 2
Now, the shared area of all three circles A, B, C=4 X the region between line ST and arc ST.
The area of the Circle A=πr2=π
The area of
SAT=12bh
=12 since b=h=r=1
So, the area between the line ST and the arc ST=14× (Area of Circle A) - Area of SAT
=14×π12=(π24)
Step 3
Area of Shaded region in Circle C=(Area of Circle C )4 (Area between the line ST and the arc ST)
=π(1)24(π24)=ππ+2=2
Therefore, Area of Shaded region in Circle C is 2.

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