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12 Feb 2023, 22:41
Kudos
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
81% (01:53) correct
19%
(02:05)
wrong
based on 73
sessions
History
Date
Time
Result
Not Attempted Yet
The circle in the diagram shown has center O and an area \(36\pi\) square feet. The area of shaded region AOB is \(6\pi\) square feet. What is the shortest possible distance between points A and B ?
A. 3
B. 6
C. \(6\sqrt{2}\)
D. 9
E. \(9\sqrt{2}\)
Attachment:
2023-02-13_09-36-57.png [ 17.05 KiB | Viewed 2310 times ]
12 Feb 2023, 23:03
Kudos
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Bunuel
The shortest distance between A and B is a straight path connecting A and B.
Let the \(\angle AOB = x^{\circ}\)
\(\frac{x^{\circ} }{ 360} * 36\pi = 6\pi\)
\(x = 60^{\circ}\)
Radius of the circle =
\(\pi * r^2 = 36\pi\)
r = 6
Method 1
Drop a perpendicular from O to AB. Line OZ bisects AB.
- \(\angle OZB = 90\)
- \(\angle ZOB = 30\)
- \(\angle OBZ = 60\)
The triangle is a 30 - 60 - 90 triangle
ZB = 3 units
AB = 3*2 = 6 units
Method 2
In \(\triangle OAB \)
\(\angle OAB = \angle OBA\)
As \(\angle AOB = 60\), the other two angles are also 60
Hence \(\triangle AOB\) is an equilateral triangle, therefore AB = 6 units.
Option B
Attachments
Screenshot 2023-02-13 113247.jpg [ 12.69 KiB | Viewed 1761 times ]
23 Sep 2023, 08:00
Kudos
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I have solved it in an entirely different way before even looking at the actual way. Tho, i got the right answer.
But i am unsure if my way was even correct or logical. COuld u have a look & let me know if it makes sense?
area of circle = 6^2pi
therefore,
radius = 6.
That is,
length of the two arcs in the diagram = 6
now,
area of triangle = 6pi
so,
6pi = 1/2 * 6 * B
when solved,
B = 2pi
B = 2 * 3.14
B = 6.28
B -~ 6
But i am unsure if my way was even correct or logical. COuld u have a look & let me know if it makes sense?
area of circle = 6^2pi
therefore,
radius = 6.
That is,
length of the two arcs in the diagram = 6
now,
area of triangle = 6pi
so,
6pi = 1/2 * 6 * B
when solved,
B = 2pi
B = 2 * 3.14
B = 6.28
B -~ 6
