Mu Alpha Theta State 2007 Euclidean Circles

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1 Mu Alpha Theta State 2007 Euclidean Circles 1. Joe had a bet with Mr. Federer saying that if Federer can solve the following problem in one minute, Joe would be his slave for a whole month. The problem was: a circle has an area of 4r 2 π, R is the radius of the circle, and D is the diameter of the circle. 3( R + 32D + 3R) 2 2 Then what is 2R π 2R π? 7R + 5R + 96D A) 360 B) r 2 + D 3 π C) r D) D 2. A = area of a circle with radius 2.5 B = volume of a cylinder with radius 2.2 and height 3.3 C = volume of a sphere with radius 3 Use 3.14 as π and find A + 2B 3C (round to the nearest tenth) A) B) C) D) There are 9 circles of equal radius inscribed in a 6 by 6 square. Assume that the circles are of maximum possible radius. What is the area outside the circles but inside the square? A) 36 36π B) 36 9π C) 36 D) 9π 4. A circle is inscribed in a right triangle with legs 3 and 4. Find the area between the triangle and the inscribed circle. A) 6 π/2 B) 6 C) 3 D) What s the value of X in degrees in the figure below? A) 3 B) 6 C) 4.5 D) 7 Error!

2 6. A triangle is inscribed within a circle. The circumference of the circle is divided into three arcs: X, Y, and Z. If one angle of the triangle is 64 and a second angle of the triangle is 36, what is the sum of the degrees in X, Y and Z? A) 720 B) 480 C) 90 D) What is the value of x in the figure below? A).41 B).42 C).43 D) A triangle with sides 60, 25, 65 is inscribed inside a circle. What is the area of the circle? 4225 A) 3600π B) 225π C) π D) π 3 9. A plane passes through a sphere 3 m away from the center, if the radius is 500 cm then what is the area of the circle that is the intersection of the plane and the sphere? A) 16π cm B) 16π m C) 1600π m D) 9π m 10. Two circles are concentric and a chord of the larger circle is tangent to the smaller circle and is equal to 10. What is the area of the annulus? A) 250π B) π C) 100π D) 25π 11. A circle is inscribed in an equilateral triangle of height 9. What is the circumference of the circle? A) 10π B) 6π C) 12π D) 18π

3 12. A circle is circumscribed about an equilateral triangle. If the side of the triangle is 2, what is the area of the circle? A) 3π B) π C) 4π D) 9π 13. A track field is made up of a rectangle, measuring 8 x 4, and 2 semicircles each attached to the shortest side of the rectangle. What is the area of the field? A) 4π + 32 B) 46π + 32 C) π D) 4π What is the perimeter of the figure in the above problem? A) 4π + 16 B) 4π + 36 C) 46π + 36 D) 6π Two circles have radii in the ratio of 2 to 5. What is the ratio of their area? A) 8 to 125 B) 1 to 1 C) 4 to 25 D) 2 to What is the area of a circle with diameter of 8? A) 64π B) no such area C) 36π D) 8π 17. A wheel with radius of x ft makes 264 revolutions rolling down a road and travels 2π miles. What is the radius of the wheel in feet? A) 20 B) 20π C) 2 D) π 18. How many circles can be drawn so that each circle is externally tangent to all the other circles? A) 3 B) 4 C) 6 D) If a circle s radius is doubled, by how much does its area increase? A) 1 B) 4 C) 6 D) Jose bought 200m of fencing wire and plans to make the biggest possible area with it. What is the area? A) 50 B) 2500 C) 200 D) (x a) 2 + (y b) 2 = r 2 is the standard equation of a circle where a is a x value of the center of a circle and b is the y value of the center of the circle and r is the radius of the circle. What is the area of a circle with equations of x 2 + y 2 + 4y = 12? A) 144π B) 4π C) 16π D) 12π

4 22. How many different arrangements are there for the word circle? A) 100 B) 520 C) 720 D) What is the volume of a sphere whose radius is equal to the height of a cone with radius of 3 and volume of 27π? A) 81π B) 972π C) 900π D) 243π 24. If a regular hexagon with edge length of 6 is inscribed in a circle what is the area of the circle? (Use 3.14 as π) A) B) 9.42 C) D) What is the radius, R, of circle O below if AB is tangent to the circle? Error! A) 4 B) 24 C) 20 D) What is the smaller angle between the hour hand and the minute hand at 3:43 pm? A) 180 B) 160 C) 146 D) Find the length of a common internal tangent segment of 2 circles with radii 4 and 12 whose centers are 20 units apart. A) 4 2 B) 4 3 C) 4 D) If the endpoints of the diameter of a circle are (2, 4) and (0, 0), what is the equation of the circle in standard form? A) (x 1) 2 + (y 1) 2 = 5 B) (x 2) 2 + (y 4) 2 = 1 C) x 2 + y 2 = 5 D) x 2 + y 2 = 1

5 29. What is the value of angle x in the circle O below if angle DOC is 79 degrees? A) 108 B) 129 C) 50 D) If there is a hexagon inscribed inside a circle of radius 6 and there is another hexagon circumscribed about the same circle, what is the ratio of the lengths of the outer hexagon to the inner hexagon? A) 4 3 : 6 B) 2 3 : 6 C) 2 3 : 3 D) 2 : 1 E)NOTA 31. What is the value of angle x in the figure below?error! A) 6 B) 12 C) 99 D) 135

6 32. If a belt is run through two pulleys of radius 1 and 6 and the centers of the two pulleys are apart by a length of 10, what is the length of the pulley? Error! A) B) π/4 C) π/2 D) π/3 33. There are 3 circles inscribed inside each other as shown below, so that the center of the largest circle lies on the circumference of the second largest circle and the center of the second largest lies on the circumference of the smallest circle. If the radius of the smallest circle is 1 then what is the area of the largest circle? A) 16π B) π C) 4π D) 36π 34. A right triangle ABC, with right angle on C, is inscribed inside a semicircle so that the largest side lies on the diameter and angle B is 30. If the radius of the semicircle is 5 then what is the area of the region inside the semicircle but outside of the triangle? A) 25π 25 3 B) 25π π C) 2 3 D) 25π

7 35. Find length AB in the figure below, if the radii of the circles are 1 and 2 and the distance between the centers is 8. Error! A) 8.2 B) 7 C) 10 D) What is the value of x in the figure blow? x + 2y 4x 4y x y 30 4x 3y A) 100 B) 60 C) 70 D) What is the circumference of a circle with area of 25? A) 5π B) 100π C) 25π D) 10π

8 38. Find ZV in the figure below if it is tangent to the circle O with radius 5. Z is the point where the line VZ is tangent to the circle O and the length of OV is 20. Assume OZ is perpendicular to VZ. A) 10 5 B) 15 5 C) 5 15 D) What is the definition of a circle? A) A one sided figure B) No such definition C) An object with no sides D) A locus of points equidistance from a common point 40. Which Greek mathematician developed a fairly accurate estimate for the circumference of the earth and a tool for determining the prime numbers between 1 and 100? A) Archimedes B) Aristarchus C) Eratosthenes D) Euclid

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