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If a rectangle has perimeter of 20 and a diagonal with length 9, what

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If a rectangle has perimeter of 20 and a diagonal with length 9, what  [#permalink]

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New post 23 Oct 2018, 03:30
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A
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E

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Question Stats:

67% (01:36) correct 33% (02:05) wrong based on 76 sessions

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If a rectangle has perimeter of 20 and a diagonal with length 9, what is the area of the rectangle?

A. 7.5
B. 8
C. 8.5
D. 9
E. 9.5
Spoiler: OA

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Re: If a rectangle has perimeter of 20 and a diagonal with length 9, what  [#permalink]

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New post 23 Oct 2018, 03:47
Bunuel wrote:
If a rectangle has perimeter of 20 and a diagonal with length 9, what is the area of the rectangle?

A. 7.5
B. 8
C. 8.5
D. 9
E. 9.5


Rectangle perimeter = 2*(L+W)

Area = L*W

20 = 2 (L+W)

L+W = 10

Given we have the diagonal length = 9

by Pythagorean rule we know \(D^2 = L^2 + W^2\)

\(81 = L^2 + W^2\)

\((L+W)^2 = L^2 + 2LW + W^2\)

Given we know \(L+W = 10\) then \((L+W)^2 = 100\)

\(100 = 81 + 2LW\)

\(2LW = 19\)

\(LW = \frac{19}{2} = 9.5\)

Answer choice E
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If a rectangle has perimeter of 20 and a diagonal with length 9, what  [#permalink]

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New post 23 Oct 2018, 05:33
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Any other shorter way to solve this question?
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Re: If a rectangle has perimeter of 20 and a diagonal with length 9, what  [#permalink]

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New post 23 Oct 2018, 05:53
Bunuel wrote:
If a rectangle has perimeter of 20 and a diagonal with length 9, what is the area of the rectangle?

A. 7.5
B. 8
C. 8.5
D. 9
E. 9.5

\(2( L + B ) = 20\)

So, \(L + B = 10\)

\(D = \sqrt{L^2 + B^2}\)

Or, \(9 = \sqrt{L^2 + B^2}\)

Or, \(L^2 + B^2 = 81\)

\(( L + B)^2 = L^2 + B^2 +2LB\)

Or, \(10^2 = 81 + 2LB\)

Or, \(2LB = 19\)

So, \(LB = 9.50\) , Answer Area must be (E) 9.50
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Re: If a rectangle has perimeter of 20 and a diagonal with length 9, what  [#permalink]

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New post 25 Oct 2018, 08:24
Bunuel wrote:
If a rectangle has perimeter of 20 and a diagonal with length 9, what is the area of the rectangle?

A. 7.5
B. 8
C. 8.5
D. 9
E. 9.5


We can let L = the length and W = the width of the rectangle. Our goal is to determine the value of LW since that is the area of the rectangle. We can create the equations:

By the Pythagorean theorem:

L^2 + W^2 = 9^2 = 81

and

by the formula for the perimeter of a rectangle:

2L + 2W = 20

L + W = 10

Squaring both sides of L + W = 10, we have:

L^2 + W^2 + 2LW = 100

Substituting, we have:

81 + 2LW = 100

2LW = 19

LW = 9.5

Answer: E
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Re: If a rectangle has perimeter of 20 and a diagonal with length 9, what  [#permalink]

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New post 26 Oct 2018, 23:31
One question: When I solved this question by quadratic equation - (l^2 + (10-l)^2) =81, I got l=(20+/-Sq.root (248))/(4).
Which results in l~= 9 or 9.1 and w~=1.

Please help me understand where I went wrong?
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Re: If a rectangle has perimeter of 20 and a diagonal with length 9, what &nbs [#permalink] 26 Oct 2018, 23:31
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