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In the figure shown above, line segment QR has length 12 and

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Carcass
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In the figure shown above, line segment QR has length 12 and [#permalink] New post   22 Jul 2020, 02:19
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In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS?

(A) 144
(B) 216
(C) 324
(D) 360
(E) 396
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Re: In the figure shown above, line segment QR has length 12 and [#permalink] New post   22 Jul 2020, 02:20
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Post A Detailed Correct Solution For The Above Questions And Get A Kudos.
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Re: In the figure shown above, line segment QR has length 12 and [#permalink] New post   22 Jul 2020, 07:10
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Carcass wrote:
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In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS?

(A) 144
(B) 216
(C) 324
(D) 360
(E) 396
Show: ::
Attachment:
Picture 1.png


Add the given information...
Image


Let x = the length of each side of square MPQT
Image

GIVEN: The area of rectangular region MPRS is 540

In other words: (base)(height) = 540
Substitute values to get: (x + 12)(x) = 540
Expand: x² + 12x = 540
Set equal to zero: x² + 12x - 540 = 0
Factor: (x + 30)(x - 18) = 0
So, EITHER x = -30 OR x = 18

Since the length cannot be negative, it must be the case that x = 18

What is the area of rectangular region TQRS?
Area = (base)(height) = (12)(18) = 216

Answer: B

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Re: In the figure shown above, line segment QR has length 12 and [#permalink] New post   24 May 2024, 10:59
how did you determine the height as 12, wouldn't the height be different from 12 as it is QR, and it is stated that only the MPQT is a square. So, we know that QR and TS can be 12, and with x that would make it 12+18, and that would be our base?
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Re: In the figure shown above, line segment QR has length 12 and [#permalink] New post   24 May 2024, 11:21
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The area of MPRS = the area of MPQT + the area of TQRS.

540 = x^2 + 12x --> x = 18.

The area of TQRS = 12*18 = 216.

Answer: B.
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Re: In the figure shown above, line segment QR has length 12 and [#permalink] New post   24 May 2024, 11:21
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x^2 + 12x - 540 = 0

We try to split 540 into two factors such that their difference is 12
540 = 54*10 = 18*3*10
You can see immediately that the 2 factors will be 18 and 30.

x^2 + 30x - 18x -540 = 0
(x + 30)(x - 18) = 0
x = 18
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Re: In the figure shown above, line segment QR has length 12 and [#permalink] New post   24 May 2024, 11:22
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Another approach

Area of rectangle.........Side QT.......Area of square.....Total area of the figure
(A) 144
(B) 216
(C) 324....................324/12=27.... 27*27 =729........... 729 + 324 = 1053
(D) 360
(E) 396

The total area was supposed to be 540 however it is 1053 hence we move up (go for smaller values). Now we know that answer is either A or B so we try any of them.


Area of rectangle.........Side QT.......Area of square.....Total area of the figure
(A) 144
(B) 216....................216/12 =18.....18*18 =324.......... 324 + 216 =540 (BINGO!)
(C) 324....................324/12=27.... 27*27 =729........... 729 + 324 = 1053
(D) 360
(E) 396

Answer is B.
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Re: In the figure shown above, line segment QR has length 12 and [#permalink]
24 May 2024, 11:22
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