Geometry Exam 2009 University of Houston Math Contest. Name: School: There is no penalty for guessing.

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1 009 Math ontest Name: School: Please read the questions carefully and give a clear indication of your answer on each question. There is no penalty for guessing. Judges will use written comments and/or calculations to settle ties. Good luck.

2 High School Mathematics ontest Spring 009 irections: You have 60 minutes to complete this exam. alculators are not permitted. hoose the correct answer for each question, and write the letter (,,, or ) corresponding to that answer in the blank to the right of each question. There is no penalty for guessing. In the case of a tie, students work will be used to determine the winner so show all work clearly on either your exam or on scrap paper. Write your name and school on top of each page in the blanks provided. Note: Geometric figures in the problems may not be drawn to scale. 1. ind the center and radius of the circle defined by the equation ( x ) ( y ) = 9. enter (, 7), Radius () enter (,7), Radius 9 () enter (,7), Radius enter (, 7), Radius 9 enter (,7), Radius If a circle has a circumference of 16π inches, find its area.. 8π in () 16π in () π in 64π in 56π in. W hat can you conclude about about Δ and Δ, given that,, and? () () The two triangles are similar. The two triangles are congruent. The two triangles are both scalene The two triangles are both acute. There is not enough information to draw a conclusion.. 4. ind the ratio of the edge of a cube to the cube s diagonal. () () 1 4.

3 5. Given that the statement If p, then q is false, which of the following statements must always be correct? I. If not p, then not q is false. II. If p, then not q is true. III. If q, then p is true. IV. If not q, then not p is false. 5. () () I and II only III only I and III only IV only II and IV only 6. RT is an altitude of Δ QRS, and m QRS = 90. If ST = 5 and QT = 7, find the length of RS. R 5 () () 1 74 Q T S 6 7. chord of length 4 cm is drawn in a circle of radius 1 cm. ind the distance from the chord to the center of the circle cm () 11 cm () 1 cm 5 cm 1 cm 8. ind the measure of an angle satisfying the following: Six times the complement of an angle is five less than the supplement of the angle. 5 () 71 () High School Mathematics ontest Spring 009

4 9. Isosceles triangle TRI has base angles T and I and median RM. If the perimeter of Δ TRI is 54 inches and TM = 4 inches, find the length of TR. 8 inches () 8 inches () inches 9. 5 inches 46 inches 10. is the median of trapezoid, with measures listed as below. ind the length of. 5 () 1 () x + 8 x + 7 5x In the circle below, chords and intersect at point. If m = 7 and m = 55, find m. () () o o Let represent the line which passes through the point ( 8, 9) and is perpendicular to the line y = x+ 6. ind the x-intercept of () 9 5 () High School Mathematics ontest Spring 009

5 1. spherical balloon has volume 40π in. If 0% more air is blown into the balloon (and it still retains its spherical shape), find the radius of the enlarged balloon in. () 6 in. () 6 in. 6 in in ind the area of a hexagon with apothem 6 inches in () 7 in () 7 in 16 in 144 in 15. elow is a net that can be used to create a solid, formed by three congruent rectangles and two congruent equilateral triangles. How many vertices and edges does the solid have? 10 vertices and 14 edges () 18 vertices and 18 edges () 6 vertices and 14 edges 9 vertices and 9 edges 6 vertices and 9 edges construction can be found below with all steps shown except for any final straightedge marking(s). Which of the following constructions is being performed? () () ngle bisector quilateral triangle Segment bisector Line through a point parallel to a given line onstruct a circle given points 16. High School Mathematics ontest Spring 009

6 17. In spherical geometry, a line is defined to be a great circle of the surface of a sphere. Two lines are parallel in spherical geometry if they never intersect. Given line on a sphere along with the above definitions, how many lines on the sphere are parallel to line? () 1 () 4 n infinite number 18. In the diagram below, ind the value of x.., m 1= ( x 7 ), m = ( x), and m = ( 9x+ 19) G 1 H 18. K () 14 () Given regular octagon GH, remove vertex G and use the remaining vertices to form heptagon H. ind the measure of. () () High School Mathematics ontest Spring 009

7 0. is a secant of circle G and is tangent to circle G at point. If m = 10 and the diameter of circle G is 10 cm, find the length of. () () 5 cm 5 cm 10 cm 5cm 0 cm G Quadrilateral is inscribed in a circle, with m = 70. ind the measure of () 110 () an not be determined. and are angle bisectors of Δ which intersect at point. If m = 110 find the measure of. () () annot be determined.. sector with central angle 80 and radius 1 has been removed from a figure, as shown below. ind the area of the portion of the circle which remains. 110 o, () () 56π 56π + 7 π 18π 11π 1 o 80. High School Mathematics ontest Spring 009

8 4. Triangle has medians,, and which intersect in point G as shown below. Given that =, = 10, and G = 11, find the length of. 4 6 () 5 () G ircle is inscribed in right triangle with right angle, and is tangent to the circle at G. If = 1 and G = 7, find. () 7 5. () G 6. Two pyramids are similar. The lateral area of the first pyramid is 40 cm, and the lateral area of the second pyramid is 540 cm. If the larger pyramid has a volume of 810 cm, find the volume of the smaller pyramid cm () 540 cm () 60 cm 70 cm 40 cm 6. High School Mathematics ontest Spring 009

9 7. company manufactures dice in large quantities and ships them to other game companies. The manufacturer gets a cube-shaped cardboard box ready to send to a customer, and fills it with 51 dice. If the dice are stacked neatly in rows in such a way that they fill the entire volume of the box, how many dice are touching at least one face of the cardboard box? 96 () 16 () cube is shown in igure 1 below. In igure, vertices,,, and of the same cube are shown as vertices of a right triangular pyramid. If the volume of the cube is 54 cm, find the volume of the right triangular pyramid. 18 cm () 1 cm () 6 cm 9 cm 8. 7 cm igure 1 igure 9. Rectangle is shown below with = 10 and = 8. If is the midpoint of, G is the midpoint of, is a point on, and the area of Δ G is 18, find the length of. 8 () 7 () G High School Mathematics ontest Spring 009

10 0. building is made from stacked cubes, and views from three different perspectives are illustrated below. There are exactly ten stacks that are one unit high. Top View: ront View: Right View: Right ront How many cubes are needed to construct this building? 0. () 4 () Which of the following polygons will tessellate the uclidean plane? I. very pentagon II. very triangle III. very hexagon IV. very quadrilateral 1. () () II only III only IV only II and IV only None of the above. ircle and ircle have common internal tangent G which intersects at. If circle has radius 10, circle has radius 8, and 7 =, find the length of. 4 () 8 () G. High School Mathematics ontest Spring 009

11 . square pyramid is made of foam and has base edge 1 cm and height 8 cm. knife slices through the apex perpendicular to the base and through two opposite vertices of the base (cutting along one of the diagonals of the base), forming two distinct congruent polyhedra. ind the total surface area of one of the resulting polyhedra. 40 cm () ( ) cm () ( ) cm ( ) cm ( ) cm. 4. Triangle exists on a coordinate plane, and the midpoints of its sides are located at 5, 4. ind the area of triangle. the points (, 1 ), (, 7 ), and ( ) 10 () 15 () In the diagram below, an equilateral triangle with side length 4 inches is shown with both its inscribed and circumscribed circles. ind the area of the gray shaded region. () () 4π in 8π 4 + in 9 16π + in 9 8 8π in 9 8π in 5. N O XM (Make sure all answer blanks are filled in with the letter of the correct answer.) High School Mathematics ontest Spring 009