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In the figure above, line L is tangent to the circle, which is centere
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09 Dec 2022, 06:11

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In the figure above, line L is tangent to the circle, which is centered at the origin. The area of the shaded region is equal to the circumference of a circle with radius between 1 and 2 1/4. Which of the following could be values of x ?

Indicate all such values.

2

−3

4

−5

−8.5

−12

Re: In the figure above, line L is tangent to the circle, which is centere
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11 Dec 2022, 04:00

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OE

Use line L to make a triangle, with points at (x, 0), the 45° angle, and the origin. The angle at (x, 0) must be 45° since the sum of the angles of a triangle is 180°. Since line L is tangential to the circle and forms 45° angles with each axis, a line from the point where line L and circle intersect to the origin will form a right angle with line L. The smaller triangle formed—from (x, 0), to where line L and circle meet, to the origin—will be a 45-45-90 triangle, with two sides equal to the radius of the circle. Find the radius of the circle, and you can find x. The area of the shaded region is equal to the circumference of a circle with radius between 1 and 2 1/4. Circumference = 2πr, so the area of the shaded region is between 2π and 4.5π, which means the area of the circle in the figure is between 8π and 18π. Area = πr^2, so the radius of the circle is between \(\sqrt{8}\) and \(\sqrt{18}\), which is to say between \(2 \sqrt{2}\) and \(3 \sqrt{2}\). 45-45-90 triangles have sides of \(a-a-a \sqrt{2}\), where, in this case, a is between \(2 \sqrt{2}\) and \(3 \sqrt{2}\). So the hypotenuse of the triangle, from the origin to (x, 0), is between \(2 \sqrt{2} \sqrt{2}\) and \(3 \sqrt{2} \sqrt{2}\), and therefore between 4 and 6, so x can range from −4 to −6.

Use line L to make a triangle, with points at (x, 0), the 45° angle, and the origin. The angle at (x, 0) must be 45° since the sum of the angles of a triangle is 180°. Since line L is tangential to the circle and forms 45° angles with each axis, a line from the point where line L and circle intersect to the origin will form a right angle with line L. The smaller triangle formed—from (x, 0), to where line L and circle meet, to the origin—will be a 45-45-90 triangle, with two sides equal to the radius of the circle. Find the radius of the circle, and you can find x. The area of the shaded region is equal to the circumference of a circle with radius between 1 and 2 1/4. Circumference = 2πr, so the area of the shaded region is between 2π and 4.5π, which means the area of the circle in the figure is between 8π and 18π. Area = πr^2, so the radius of the circle is between \(\sqrt{8}\) and \(\sqrt{18}\), which is to say between \(2 \sqrt{2}\) and \(3 \sqrt{2}\). 45-45-90 triangles have sides of \(a-a-a \sqrt{2}\), where, in this case, a is between \(2 \sqrt{2}\) and \(3 \sqrt{2}\). So the hypotenuse of the triangle, from the origin to (x, 0), is between \(2 \sqrt{2} \sqrt{2}\) and \(3 \sqrt{2} \sqrt{2}\), and therefore between 4 and 6, so x can range from −4 to −6.

Re: In the figure above, line L is tangent to the circle, which is centere
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12 Aug 2024, 10:10

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Re: In the figure above, line L is tangent to the circle, which is centere [#permalink]

12 Aug 2024, 10:10
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