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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 11 Oct 2018, 03:00
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In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?


(A) \(\frac{3}{12}\)

(B) \(\frac{24}{49}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{25}{49}\)

(E) \(\frac{7}{12}\)


Attachment:
Capture (2).JPG
Capture (2).JPG [ 15.52 KiB | Viewed 1143 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 11 Oct 2018, 20:05
1
Bunuel wrote:
Image
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?


(A) \(\frac{3}{12}\)

(B) \(\frac{24}{49}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{25}{49}\)

(E) \(\frac{7}{12}\)


Attachment:
The attachment Capture (2).JPG is no longer available


So every corner makes a right angled triangle with sides 3-4-5...
And hypotenuse of this triangle is the side of inner square, so 5..

Area of shaded region = \(7^2-5^2=49-25=24\)
Total area =7^2=49

Shaded region as a fraction of total =\(\frac{24}{49}\)

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 13 Oct 2018, 18:50
Bunuel wrote:
Image
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?


(A) \(\frac{3}{12}\)

(B) \(\frac{24}{49}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{25}{49}\)

(E) \(\frac{7}{12}\)


Attachment:
Capture (2).JPG


The area of the larger square is 7^2 = 49. We see that each of the shaded regions is a right triangle that is a 3-4-5 right triangle. Thus, each of these right triangles has an area of ½ x 3 x 4 = 6, and the total area of the four triangles is 6 x 4 = 24. So the fraction of the larger square is shaded is 24/49.

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

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New post 05 Jun 2019, 11:00
chetan2u ScottTargetTestPrep
I have question ; we are given that a square is inscribed in square ; we can notice that the digonal of the inside square = 7 so side of small square ; s=7/√2
area = 49/2
so shaded region ; 49-49/2 = 49/2
ratio ; 49/2 /49 = 1/2
IMO C

why is this approach and answer wrong?



chetan2u wrote:
Bunuel wrote:
Image
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?


(A) \(\frac{3}{12}\)

(B) \(\frac{24}{49}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{25}{49}\)

(E) \(\frac{7}{12}\)


Attachment:
Capture (2).JPG


So every corner makes a right angled triangle with sides 3-4-5...
And hypotenuse of this triangle is the side of inner square, so 5..

Area of shaded region = \(7^2-5^2=49-25=24\)
Total area =7^2=49

Shaded region as a fraction of total =\(\frac{24}{49}\)

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 05 Jun 2019, 11:31
Archit3110 wrote:
chetan2u ScottTargetTestPrep
I have question ; we are given that a square is inscribed in square ; we can notice that the digonal of the inside square = 7 so side of small square ; s=7/√2
area = 49/2
so shaded region ; 49-49/2 = 49/2
ratio ; 49/2 /49 = 1/2
IMO C

why is this approach and answer wrong?



chetan2u wrote:
Bunuel wrote:
Image
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?


(A) \(\frac{3}{12}\)

(B) \(\frac{24}{49}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{25}{49}\)

(E) \(\frac{7}{12}\)


Attachment:
Capture (2).JPG


So every corner makes a right angled triangle with sides 3-4-5...
And hypotenuse of this triangle is the side of inner square, so 5..

Area of shaded region = \(7^2-5^2=49-25=24\)
Total area =7^2=49

Shaded region as a fraction of total =\(\frac{24}{49}\)

B


The highlighted part is not correct, you can not calculate the diagonal of the square that way, chetan2u s approach is IMHO the most elegant way to solve the question...
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Re: In the 7-inch square above, another square is inscribed. What fraction  [#permalink]

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New post 05 Jun 2019, 11:47
Abhishek009
Could you please share reason why won't the diagonal of small square be 7 ? Is it because figure is not drawn to scale or some other reason?

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

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New post 05 Jun 2019, 12:37
Archit3110 wrote:
Abhishek009
Could you please share reason why won't the diagonal of small square be 7 ? Is it because figure is not drawn to scale or some other reason?

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Bro , the sides of the square inscribed inside the square is 5 units and the sides of the bigger square is 7 units....
Attachment:
Capture (2).JPG
Capture (2).JPG [ 16.57 KiB | Viewed 441 times ]

Thus the diagonal of the bigger square will be 7√2 and the diagonal of the smaller square will be 5√2

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

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New post 05 Jun 2019, 13:02
Hi Archit3110 , the diagonal of the inner square would have been 7 units if in the figure, all the points, where the inner square is touching the outer square are the midpoints of the sides of outer square

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

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New post 06 Jun 2019, 17:59
Archit3110 wrote:
chetan2u ScottTargetTestPrep
I have question ; we are given that a square is inscribed in square ; we can notice that the digonal of the inside square = 7 so side of small square ; s=7/√2
area = 49/2
so shaded region ; 49-49/2 = 49/2
ratio ; 49/2 /49 = 1/2
IMO C

why is this approach and answer wrong?



chetan2u wrote:
Bunuel wrote:
Image
In the 7-inch square above, another square is inscribed. What fraction of the larger square is shaded?


(A) \(\frac{3}{12}\)

(B) \(\frac{24}{49}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{25}{49}\)

(E) \(\frac{7}{12}\)


Attachment:
Capture (2).JPG


So every corner makes a right angled triangle with sides 3-4-5...
And hypotenuse of this triangle is the side of inner square, so 5..

Area of shaded region = \(7^2-5^2=49-25=24\)
Total area =7^2=49

Shaded region as a fraction of total =\(\frac{24}{49}\)

B


Yes, your assumption is wrong. From the picture, you can actually see that the diagonal of the inscribed square is slanted, that is, it’s actually longer than a side of the circumscribed square. The only way the diagonal of the inscribed square is equal to the side length of the circumscribed square is when the vertices of the inscribed square are the midpoints of the sides of the circumscribed square. However, that is not the case here. You can see that the vertices of the inscribed square are a little off from the midpoints of the sides of the circumscribed square.
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Re: In the 7-inch square above, another square is inscribed. What fraction   [#permalink] 06 Jun 2019, 17:59
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