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 in inches? (Note: Figure not drawn to scale.)
A. 1 B. 1.5 C. 2 D. 2.5 E. 3
(B) Triangle XYZ is an isosceles right triangle with hypotenuse of 3√2. The property of an isosceles right triangle is that the hypotenuse is √2 times greater than any other side. Thus, 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).
---------- I don't understand what 3√2 means, how has this been simplified? How do you extract that each side is 3 from this length?
Because we are working with a square, the triangle XZY is a 45-45-90 isosceles right triangle. And, we know from our study of classical right triangles, the hypotenuse of such a triangle is equal to √2 times the length of one of the legs. The length of the leg happens to be the length of the side of the square. So the length of each of the sides of the square is equal to 3.
Now notice that the diameter of the circle is equal to the length of one of the sides of the square. Convince yourself by drawing a vertical like through the circle that passes its center. So, the diameter is equal to 3. The radius is equal to half the diameter, so the radius is equal to 1.5.
We know that 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 hypotenuse is (√2 × 3), we clearly see that the leg is 3.
I fail to understand how the leap from 3√2 to 3 is achieved. To my mind the symbol 3√2 means the square root of 2 (1.41) multipied by 3 to give 4.24.
The answer to your question is right above in the previous posts.
Note, that your understanding of the symbol 3√2 as 3 × √2 is correct. Though you should understand that 1.41 is the approximate value of √2. Besides, usage of such approximation in this question will definitely make the solution longer. It's better to use square roots when we deal with right triangles.
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?
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