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What is the longest straight line that can be inscribed in a

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What is the longest straight line that can be inscribed in a [#permalink] New post 21 May 2013, 01:09
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What is the longest straight line that can be inscribed in a rectangle with a perimeter of 16 and sides of integer length?

A. 4 root 2
B. 6
C. 7
D. 5 root 2
E. 7 root 2

Grockit! please Explain
[Reveal] Spoiler: OA

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Re: What is the longest straight line that can be inscribed in a [#permalink] New post 21 May 2013, 01:18
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fozzzy wrote:
What is the longest straight line that can be inscribed in a rectangle with a perimeter of 16 and sides of integer length?

A. 4 root 2
B. 6
C. 7
D. 5 root 2
E. 7 root 2

Grockit! please Explain


Basically we are asked to maximize the length of the diagonal in a rectangle with a perimeter of 16.

Say a and b are the lengths of the sides, then we have that 2a + 2b = 16 --> a + b = 8.

We need to maximize d=\sqrt{a^2+b^2}. Since a and b are integers then d_{max} is obtained when a=1 and b=7 or vise-versa (by trial and error) --> d=\sqrt{a^2+b^2}=\sqrt{50}=5\sqrt{2}.

Answer: D.

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Re: What is the longest straight line that can be inscribed in a [#permalink] New post 21 May 2013, 01:24
Longest straight line that can be inscribed in a rectangle = the diagonal of that rectangle.
Longest diagonal can be obtained when one side of the rectangle has shortest possible length and the other side has the longest possible length when perimeter is constant.

In this case, as the sides are integers, shortest side can be 1 and longest side can be 7.

Then, diagonal of the rectangle = \sqrt{(1^2 + 7^2)} = \sqrt{50} = 5\sqrt{2}

Answer is D.
Re: What is the longest straight line that can be inscribed in a   [#permalink] 21 May 2013, 01:24
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