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Author:  questioner [ Sat Apr 14, 2012 6:15 am ]
Post subject:  GRE Geometry (Select One)

Image

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

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

(B) 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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I am confused how right away in the answer we get the sides of the square to be three.


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Author:  Gennadiy [ Sat Apr 14, 2012 6:33 am ]
Post subject:  Re: t.2, s.2, qt.1: Geometry, a circle inscribed in a square

We know that the triangle ZXY is isosceles and right. It is a property of an isosceles right triangle that its legs are equal and the hypotenuse is √2 times greater. The alternative method in the explanation proves it.

Therefore, since we know that the hypotenuse is (√2 × 3), we clearly see that the leg is 3.

Author:  questioner [ Sat Apr 14, 2012 6:34 am ]
Post subject:  Re: t.2, s.2, qt.1: Geometry, a circle inscribed in a square

The explanation states that each leg is equal to 3 inches but the question states that leg XZ is 3√2. Since it's a square, all legs are 3√2. How can each leg of the triangle (or side of the square) be 3 inches if the problem states that it's 3√2?

Author:  Gennadiy [ Sat Apr 14, 2012 6:35 am ]
Post subject:  Re: t.2, s.2, qt.1: Geometry, a circle inscribed in a square

Quote:
the question states that leg XZ is 3√2. Since it's a square, all legs are 3√2.
XZ is NOT a side of the square. XZ is its diagonal. The sides are ZY, YX, XW, WZ.
You may find above the explanations of why the sides are 3.

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