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In the figure above, a circle with center O is inscribed in the square

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Bunuel
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In the figure above, a circle with center O is inscribed in the square [#permalink] New post   26 Dec 2014, 08:31
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Tough and Tricky questions: Geometry.




In the figure above, a circle with center O is inscribed in the square WXYZ. The segment XZ has a length of 3√2 inches. What is the radius of the circle in inches?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3

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2014-12-26_1931.png
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B
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   26 Dec 2014, 10:02
The diagonale of the square has a length of \(3\sqrt{2}\) inches. Therefore, one side of the square has the length 3 inches. Since the circle in inscribed in the square, the radius must be one half the length of the square's side: 1.5 inches.

Answer B.
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   26 Dec 2014, 17:26
Bunuel wrote:

Tough and Tricky questions: Geometry.




In the figure above, a circle with center O is inscribed in the square WXYZ. The segment XZ has a length of 3√2 inches. What is the radius of the circle in inches?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3

Kudos for a correct solution.

Show Spoiler
Attachment:
2014-12-26_1931.png


XZ = 3√2

The outer figure should be square as the circle is inscribed completely inside the figure

xz^2 = xy^2 + yz^2
xz = √2xy^2 --- (since xy=yz)
3√2 = xy√2
xy = 3

xy = 2 * r


Ans B
r = 1.5
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   27 Dec 2014, 01:08
Since the diagnol of square is 3 sqrt. 2 . The side of the square will be 3 each. We as sqaure diagnol:side=x*sqrt2. where x is the side of the square.

You can verify the same through pythagoreas theorem.

Now Side of the square is diameter of the circle. hence diameter is 3. Thus radius is 1.5. Thus answer is B.
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   02 Jan 2015, 02:13
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Answer = B = 1.5

Refer diagram below:

Attachment:
2014-12-26_1931.png
2014-12-26_1931.png [ 7.01 KiB | Viewed 9788 times ]


We require to find the side of the modified square per diagram below which is the radius of the circle

\(= \sqrt{\frac{3}{\sqrt{2}} * \frac{3}{\sqrt{2}} * \frac{1}{2}} = \frac{3}{2} = 1.5\)
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   08 Jan 2015, 09:33
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Bunuel wrote:

Tough and Tricky questions: Geometry.




In the figure above, a circle with center O is inscribed in the square WXYZ. The segment XZ has a length of 3√2 inches. What is the radius of the circle in inches?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3

Kudos for a correct solution.

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Attachment:
2014-12-26_1931.png


OFFICIAL SOLUTION:

Triangle XYZ is an isosceles right triangle. The length of each leg is 3. Since the diameter of the circle is equal to the height of the square, the diameter is equal to 3. So the radius is equal to 1.5 inches.

Alternatively, since the sides XY and ZY are equal, set them equal to x. Then 2x² = (3√2)² by the Pythagorean Theorem. Solving for x: x² = 9 and so x = 3. Since x represents the side of the square or the diameter of the circle, we divide by 2 to get the radius. So, the radius must be 1.5.

The correct answer is (B).
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   30 Jul 2017, 20:01
Sorry i have question here. why is the side of the square the diameter of the circle? Also the diagonal of the square is the diameter of the circle right?
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   30 Jul 2017, 20:24
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longhaul123 wrote:
Sorry i have question here. why is the side of the square the diameter of the circle? Also the diagonal of the square is the diameter of the circle right?


If a circle is inscribed in a square, the diameter of the circle will be equal to the side of the square. Check the image below:
Attachment:
Untitled.png
Untitled.png [ 3.88 KiB | Viewed 7186 times ]
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   01 Aug 2017, 00:58
Bunuel wrote:

Tough and Tricky questions: Geometry.




In the figure above, a circle with center O is inscribed in the square WXYZ. The segment XZ has a length of 3√2 inches. What is the radius of the circle in inches?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3

Kudos for a correct solution.

Show Spoiler
Attachment:
2014-12-26_1931.png



As the given figure is a square. Triangles XZW and XYZ are isoceles right triangles, which is a 45-45-90 triangle. The ratio of the sides of this kind of triangle is 1: 1: \(\sqrt{2}\).
Given that the diagonal, which is hypotenuse for both the triangles, is 3 \(\sqrt{2}\)
Therefore, the sides of the square= 3
When circle is inscribed in a square, diameter of the circle = sides of the square (= 3 here).
Thus, radius= 3/2= 1.5
Answer: B
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   01 Aug 2017, 17:21
Expert Reply
Bunuel wrote:

Tough and Tricky questions: Geometry.




In the figure above, a circle with center O is inscribed in the square WXYZ. The segment XZ has a length of 3√2 inches. What is the radius of the circle in inches?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3


We see that 3√2 = the diagonal of the square; thus:

3√2 = side√2

3 = side of square

Since the side of the square also equals the diameter of the circle, the radius of the circle is 1.5.

Answer: B
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Re: In the figure above, a circle with center O is inscribed in the square [#permalink] New post   15 Jun 2021, 04:49
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