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Problem of the Week
Problem E and Solution
Circle This
Problem
M ON is a sector of a circle with radius ON which is 6 cm long. If ∠M ON = 60◦ , determine
the radius of the circle which passes through the points M , N , and O.
Solution
Let C be the centre of the circle that passes through M , N , and O. Then CM ,
CN , and CO are radii. Therefore, CM = CN = CO = r.
In 4CM O and 4CN O, CM = CN , CO is common and OM = ON .
Therefore, 4CM O ∼
= 4CN O and it follows that ∠COM = ∠CON . But
◦
∠M ON = 60 . Therefore, ∠COM = ∠CON = 30◦ .
In 4CM O, CM = CO = r and 4CM O is isosceles. Therefore,
∠CM O = ∠COM = 30◦ and ∠M CO = 180◦ − 30◦ − 30◦ = 120◦ .
Method 1: Using the sine law,
OM
CM
=
sin (∠COM )
sin (∠M CO)
r
6
=
sin 30◦
sin 120◦
6
r =
× sin 30◦
◦
sin 120
6
1
r = √ ×
3
2
2
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1
2
r = 6× √ ×
3√ 2
6
3
r = √ ×√
3
√3
r = 2 3 cm
√
The radius of the circle that passes through M , N , and O is 2 3 cm.
Method 2: Using the cosine law,
CM 2
r2
12r cos 30◦
r cos √
30◦
3
r×
√2
r× 3
√
√
r× 3× 3
3r
r
=
=
=
=
CO2 + M O2 − 2 × CO × M O × cos (∠COM )
r2 + 62 − 2(6)(r) cos 30◦
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3
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= 3
=
=
=
=
6
√
6× 3
√
6 3
√
2 3 cm
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√
The radius of the circle that passes through M , N , and O is 2 3 cm.
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