A geometry problem by Majed Musleh
Geometry
Level
pending
In the above image if
Find the area of
100(Pi-2)
10(Pi-2)
20(Pi-2)
50(Pi-2)
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1 solution
First i will write the equations of the three circles A − x 2 + y 2 = 4 0 0 B − ( x − 1 0 ) 2 + y 2 = 1 0 0 C − x 2 + ( y − 1 0 ) 2 = 1 0 0 Now the area L = G how? G = t h e a r e a o f q u a r t e r A B C − t h e a r e a o f t h e s e m i c i r c l e B C − t h e s e m i c i r c l e A C + L we add L because we substract it twice when we substract the two semicircles G = π 4 2 0 2 − π 2 1 0 2 − π 2 1 0 2 + L do the calculation ... = > G = L now we need to find the points of intersection from eq. B and eq. C x 2 − 2 0 x + 1 0 0 + y 2 = x 2 + y 2 − 2 0 y + 1 0 0 => x = y =>substitute in eq. B we get y = 0 = > x = 0 o r y = 1 0 = > x = 1 0 the points of intersection leads us to drow the square with side length 1 0 as shown in the photo
now the area of
G
=
t
h
e
a
r
e
a
o
f
q
u
a
r
t
e
r
A
B
C
−
t
h
e
a
r
e
a
o
f
t
h
e
s
q
u
a
r
e
−
2
t
h
e
a
r
e
a
o
f
t
h
e
q
u
a
r
t
e
r
o
f
c
i
r
c
l
e
w
i
t
h
r
=
1
0
G
=
π
4
2
0
2
−
1
0
2
−
2
π
4
1
0
2
=
5
0
π
−
1
0
0
=
5
0
(
π
−
2
)
Finally
L
+
G
=
2
∗
5
0
(
π
−
2
)
=
1
0
0
(
π
−
2
)