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

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Director
Joined: 29 Nov 2012
Posts: 818
What is the longest straight line that can be inscribed in a [#permalink]

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21 May 2013, 02:09
1
4
00:00

Difficulty:

35% (medium)

Question Stats:

70% (01:15) correct 30% (01:30) wrong based on 213 sessions

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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

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Math Expert
Joined: 02 Sep 2009
Posts: 47047
Re: What is the longest straight line that can be inscribed in a [#permalink]

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21 May 2013, 02:18
4
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

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) --> $$d=\sqrt{a^2+b^2}=\sqrt{50}=5\sqrt{2}$$.

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Posts: 378
Re: What is the longest straight line that can be inscribed in a [#permalink]

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21 May 2013, 02:24
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 = $$\sqrt{(1^2 + 7^2)} = \sqrt{50} = 5\sqrt{2}$$

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Re: What is the longest straight line that can be inscribed in a [#permalink]

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02 Aug 2017, 08:54
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Re: What is the longest straight line that can be inscribed in a   [#permalink] 02 Aug 2017, 08:54
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