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A rectangle is inscribed in a circle of radius r
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Updated on: 10 Nov 2020, 10:55
Updated on: 10 Nov 2020, 10:55
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Question Stats:
56% (01:36) correct
43% (01:39) wrong based on 37 sessions
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A rectangle is inscribed in a circle of radius r. If the rectangle is not a square, which of the following could be the perimeter of the rectangle?
A. \(2r \sqrt 3\)
B. \(2r (\sqrt 3 + 1)\)
C. \(4r \sqrt 2\)
D. \(4r \sqrt 3\)
E. \(4r (\sqrt 3 + 1)\)
A. \(2r \sqrt 3\)
B. \(2r (\sqrt 3 + 1)\)
C. \(4r \sqrt 2\)
D. \(4r \sqrt 3\)
E. \(4r (\sqrt 3 + 1)\)
Originally posted by daina1323031 on 10 Nov 2020, 10:42.
Last edited by Carcass on 10 Nov 2020, 10:55, edited 2 times in total.
Last edited by Carcass on 10 Nov 2020, 10:55, edited 2 times in total.
Formatted
Re: A rectangle is inscribed in a circle of radius r
[#permalink]
10 Nov 2020, 10:54
10 Nov 2020, 10:54
Expert Reply
Please Sir.
When you post a question, identify it with the words of the stem and NOT simply geometry
Thank you for your collaboration
When you post a question, identify it with the words of the stem and NOT simply geometry
Thank you for your collaboration
Re: A rectangle is inscribed in a circle of radius r
[#permalink]
10 Nov 2020, 10:56
10 Nov 2020, 10:56
1
Expert Reply
Since the radius of the circle is r, the diameter must be 2r. Now imagine the rectangle. The diameter must be the hypotenuse of the right angled triangle of the rectangle. Say, if its sides are a and b,
\((2r)^2 = a^2 + b^2 = 4r^2\)
So when you square a and b and sum them, you should get 4r^2
The given options are the perimeter of the rectangle i.e. 2(a+b). So I ignore 2 of the options and try to split the leftover into a and b.
The obvious first choices are options (B) and (E) since we can see that we can split the sum into 3 and 1.
\(a + b = r\sqrt{3} + r\)
\((\sqrt{3}r)^2 + r^2 = 4r^2\)
The answer is (B)
\((2r)^2 = a^2 + b^2 = 4r^2\)
So when you square a and b and sum them, you should get 4r^2
The given options are the perimeter of the rectangle i.e. 2(a+b). So I ignore 2 of the options and try to split the leftover into a and b.
The obvious first choices are options (B) and (E) since we can see that we can split the sum into 3 and 1.
\(a + b = r\sqrt{3} + r\)
\((\sqrt{3}r)^2 + r^2 = 4r^2\)
The answer is (B)
