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17 Jan 2022, 03:15
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
69% (02:32) correct
31%
(02:01)
wrong
based on 13
sessions
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Not Attempted Yet
PM and PN are tangent to circle O at M and N respectively; \(\angle MON=120°\) and \(OM = ON = 5\). What is the perimeter of the shaded region?
(A) \(10+10\pi\)
(B) \(5\sqrt{3}+10\pi\)
(C) \(5\sqrt{3}+\frac{10\pi}{3}\)
(D) \(10\sqrt{3}+\frac{10\pi}{3}\)
(E) \(\sqrt{3}+10\pi\)
Are You Up For the Challenge: 700 Level Questions
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18 Jan 2022, 15:43
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Lets start by finding curved side MN.
To do this we need to find a sector of a circle's circumference we need to multiply the circumference by whatever fraction the internal angle is. So:
The angle is: 120/360 = 1/3
We are given the radius r=5
Circumference of circle = 2πr = 10π
Curved side MN = 10π * 1/3 = 10π/3
Now to find the sides MP and NP which are equal.
A tangent intersected by the radius (OM or ON) will give us a right angle.
Now if we connect P to O with a line we have two right triangles:
One angle is 90.
Line PO bisects the 120 degree angle ===> Other angle 60.
The final angle we find by 180-90-60= 30
===> We have two 30:60:90 triangles. We know the short leg is the radius as it is opposite the smallest angle.
===> Therefore the long leg (MP or NP) = r square root of 3 = 5 sqrt 3
Now we multiply that value by 2 to account for both sides (MP & NP). And add that value to curved side MN.
===> 2 * (5 sqrt 3) + 10π/3 = 10 sqrt 3 + 10π/3
===> D
To do this we need to find a sector of a circle's circumference we need to multiply the circumference by whatever fraction the internal angle is. So:
The angle is: 120/360 = 1/3
We are given the radius r=5
Circumference of circle = 2πr = 10π
Curved side MN = 10π * 1/3 = 10π/3
Now to find the sides MP and NP which are equal.
A tangent intersected by the radius (OM or ON) will give us a right angle.
Now if we connect P to O with a line we have two right triangles:
One angle is 90.
Line PO bisects the 120 degree angle ===> Other angle 60.
The final angle we find by 180-90-60= 30
===> We have two 30:60:90 triangles. We know the short leg is the radius as it is opposite the smallest angle.
===> Therefore the long leg (MP or NP) = r square root of 3 = 5 sqrt 3
Now we multiply that value by 2 to account for both sides (MP & NP). And add that value to curved side MN.
===> 2 * (5 sqrt 3) + 10π/3 = 10 sqrt 3 + 10π/3
===> D
