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In the 7-inch square above, another square is inscribed. What fraction

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In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 24 Nov 2014, 21:42
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Question Stats:

73% (01:38) correct 27% (01:51) wrong based on 343 sessions

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In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?( picture is on the attachment)

(A) 3/12
(B) 24/49
(C) 1/2
(D) 25/49
(E) 7/12

my ans is c but OA is b, what wil be right one?

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1.jpg
1.jpg [ 21.29 KiB | Viewed 5201 times ]

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In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 08 Sep 2017, 12:20
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anik89 wrote:
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?( picture is on the attachment)

(A) 3/12
(B) 24/49
(C) 1/2
(D) 25/49
(E) 7/12


The first thing we need to recognize is that the 4 SHADED triangles are identical. Here's why:

If we focus on one triangle and let x and y represent the 2 angles that are not 90 degrees, we get the following:
Image
From this, we can see that x + y + 90° = 180°

So, when we examine another right triangle (this time in green)....
Image
...we can see that the angle represented by ? must be equal to x, since we already know that x + y + 90° = 180°

So, we'll add that information to the diagram...
Image

Using similar logic, we can see that the 3rd angle in the green triangle must be equal to y, since we know that x + y + 90° = 180°
Image

Using the exact same process, we can show that all 4 shaded triangles have the same 3 angles: x, y and 90°
We're told that the white quadrilateral in the middle of the diagram is a SQUARE, so we can see that the hypotenuse in each of the SHADED triangles has the same length.
This means that all 4 SHADED triangles are identical....
Image

So, the area of each SHADED triangle = (base)(height)/2 = (4)(3)/2 =12/2 = 6
So, the TOTAL area of all 4 SHADED triangles = (4)(6) = 24

The area of the LARGER square = (7)(7) = 49

So, 24/49 of the larger square is shaded

Answer:

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Re: In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 25 Nov 2014, 02:05
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anik89 wrote:
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?( picture is on the attachment)

(A) 3/12
(B) 24/49
(C) 1/2
(D) 25/49
(E) 7/12

my ans is c but OA is b, what wil be right one?


each shaded represents a right angle triangle with 3,4 as its perpendicular sides. area of one of these right angle triangle = \(\frac{3.4}{2}\) \(= 6\)

thus total area of the shaded region \(= 4.6 =24\)

and the required fraction = (area of shaded region)/(area of the larger square) = \(\frac {24}{7^2}\) \(= 24/49\)
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Re: In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 25 Nov 2014, 07:47
anik89 wrote:
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?( picture is on the attachment)

(A) 3/12
(B) 24/49
(C) 1/2
(D) 25/49
(E) 7/12

my ans is c but OA is b, what wil be right one?


Another way to do this problem is to notice that these side lengths are going to give us a special right triangle of 3-4-5. Since we know both the outer and inner objects are squares, we know that the side ratios are going to be the same on all sides. This means all of those outer triangles will be 3-4-5 right triangles.

Since this is the case, we know the inner square is a 5x5, with area of 25 square units. Taking the side length of the outer square, we know it is 7x7 or 49 square units. This means the inner square is 25/49 the size of the outer square. Subtracting 25 from 49 gives us the ratio of the shaded region, 24/49.

So the answer is B.
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Re: In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 25 Nov 2014, 18:04
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Answer = (B) \(\frac{24}{49}\)

There are 4 triangles with base, height = 4, 3

Total shaded region area \(= 4 * \frac{1}{2} * 4 * 3 = 24\)

Square area \(= 7^2 = 49\)

Fraction \(= \frac{24}{49}\)
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Re: In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 25 Nov 2014, 18:11
anik89 wrote:
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?( picture is on the attachment)

(A) 3/12
(B) 24/49
(C) 1/2
(D) 25/49
(E) 7/12

my ans is c but OA is b, what wil be right one?


As far as your calculation is concerned, answer would be C (1/2) when the small square is inscribed at mid points of the 7 inch square (per diagram below)

Attachment:
sq.png
sq.png [ 3.55 KiB | Viewed 5044 times ]

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Re: In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 21 Jan 2016, 15:19
Hi All,

The error that was made by the original poster was in assuming that the answer to THIS prompt is the same as the answer to a slightly different version of this prompt. IF the interior square had its corners at the exact midpoint of each of the 4 sides of the larger square, THEN exactly HALF of the larger square would be shaded. Since that's NOT the case in this prompt (the 4 corners are NOT at the exact midpoints), then the shaded area will be LESS than half.

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In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 31 Jul 2016, 20:29
1
anik89 wrote:
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?( picture is on the attachment)

(A) 3/12
(B) 24/49
(C) 1/2
(D) 25/49
(E) 7/12

my ans is c but OA is b, what wil be right one?


Outer square area = \(7^2 = 49\)

Inner square area = \(\frac{Diagonal^2}{2}\)= \(\frac{7^2}{2} = \frac{49}{2}\)

Total area = 49
Unshaded area (inner square) is 49/2 which is half of 49
so obviously the remaining shaded area will also be half of the total area =49/2
Shaded area = 1/2
Unshaded area = 1/2

Remember the question is asking for "What fraction of "
ANSWER is C
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Re: In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 12 Aug 2018, 07:51
anik89 wrote:
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?( picture is on the attachment)

(A) 3/12
(B) 24/49
(C) 1/2
(D) 25/49
(E) 7/12

my ans is c but OA is b, what wil be right one?


Great question. Will a Moderator please update the question to erase the highlighted part above so that we don't see the OA while working through the timers :grin:

Thanks,
Mike
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Re: In the 7-inch square above, another square is inscribed. What fraction &nbs [#permalink] 12 Aug 2018, 07:51
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