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

Question

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

Salsanousi · Answer

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

dollytaneja51 · Answer

Posted from my mobile device

Any other shorter way to solve this question?

Abhishek009 · Answer

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

ScottTargetTestPrep · Answer

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

ABS193 · Answer

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?

DmitryFarberMPrep · Answer

Quick solution:

If the perimeter=20, then 2 adjacent sides will sum to 10. Using the diagonal, we have a triangle with hypotenuse 9 and legs that sum to 10. 1 and 9 will make a hypotenuse just barely bigger than 9. 2 and 8 will make a hypotenuse closer to 8. So our actual area is between 9*1=9 and 8*2=16. That leaves only E.

If there were a larger answer, it would help to notice that 9 and 1 are very close to the actual values, but that the area increases as the sides draw closer to each other in value.

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

Prep Toolkit

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