Geometry Contest for 2010

Transcription

1 Geometry ontest for ) Two angles are complementary if their sum is ) 180 ) between 90 and 180 ) 90 ) less than 90 ) more than 180 2) If two exterior angles of a triangle are congruent, then the triangle is ) acute ) scalene ) isosceles ) obtuse ) equilateral 3) ssume that quadrilateral is a parallelogram. To completely prove is a rectangle if and only if = we need to prove: ) If is a rectangle, then =. ) If = in parallelogram, then is a rectangle. ) If is not a rectangle, then. ) If is a rectangle with =, then is a parallelogram. ) Need to prove both () and (). 4) If two parallel lines are cut by a transversal, then the ) Interior angles on the same side of the transversal are congruent. ) lternate interior angles are supplementary. ) lternate interior angles are congruent. ) orresponding angles are supplementary. ) lternate exterior angles are supplementary. 5) Which of these is the contrapositive of If I have a nickel, then I am rich? ) If I am rich, then I have a nickel. ) If I don t have a nickel, then I am not rich. ) If I am not rich, then I don t have a nickel. ) If I have a nickel, then I am rich. ) I am rich if and only if I have a nickel 6) If two distinct planes intersect, then their intersection is ) two parallel lines ) a line ) a point ) a plane ) a plane crash

2 7) How many different length line segments are there whose endpoints are on a 5 X 5 geoboard? ) 4 ) 5 ) 8 ) 9 ) 14 8) If the sides of a triangle are 20, 21, and 29 units, then the triangle is ) acute ) isosceles ) obtuse ) right ) not possible 9) The circles with centers at,, and are mutually tangent at,, and F as shown. ompute. ) 1 ) Ø ) 2 ) cannot be determined F 10) is inscribed in a circle with diameter. If m (the measure of arc ) = 30, find the measure of. ) 60 ) 150 ) 75 ) 90 ) 15 O

3 11) If and are points of tangency to the circle and is an arbitrary point on minor arc, find m + m. ) 90 ) between 90 and 180 ) 180 ) between 180 and 360 ) ) Segment is a diagonal in parallelogram. Incircles of and have points of tangency at, F, G, H, I, and J as shown. Find IJ: ) ) ) ) + I J H I J G F 13) Two semicircles are constructed in a quadrant of a circle as shown. If the diameter of the larger semicircle and the radius of the quadrant are each 8 units, find the radius of the smaller semicircle. ) ) 1 ) 2 ) ) 4

4 14) Find the area of the ring between two concentric circles if chord of the larger circle is tangent at point T of the smaller circle and = 8. ) 2π ) 8π ) 12π ) 16π ) insufficient information to solve. T 15) The three triangles in the figure are scalene. Segments,, and are all concurrent at G. Find m + m + m + m + m + m F. ) 90 ) 180 ) 270 ) 360 ) cannot be determined F G 16) Rays and bisect exterior angles of. If m = 90, find m. ) 30 ) 45 ) 60 ) 75 ) cannot be determined

5 17) Given: with = 4, = 6, = 2, and = 4. Find the area of. ) 14 ) 28 ) 56 ) 63 ) 12 18) Four triangles have sides of lengths (in cm) as given by: a) b) c) d) Which one has the largest area? ) ) ) ) ) ll four areas have the same measure. 19) regular pentagon is inside a regular hexagon and shares side as shown. Find m. ) 12 ) 20 ) 36 ) 60 ) 8 20) If is a triangle and is a triangle, find the ratio of the area of to the area of. ) 1 ) ) ) )

6 21) Which of the following does not form a regular tessellation? ) an equilateral triangle ) a regular polygon of 4 sides ) a regular pentagon ) a regular hexagon ) ll of the above form regular tessellations. 22) Let be a general convex quadrilateral whose diagonals meet at. Let F, G, H and I be the centroids of,,, and, respectively. What kind of quadrilateral must FGHI be? ) Square ) Rhombus ) Rectangle ) Parallelogram ) Trapezoid 23) How many diagonals are there in a convex hexagon? ) 3 ) 5 ) 6 ) 8 ) 9 24) If each of the dimensions of a cube is doubled to form a new cube, then what is the ratio of the volume of the original cube to the volume of the new cube? ) ) ) ) ) 25) Given an arbitrary triangle which of the following concurrency points are always collinear? ) orthocenter, incenter, centroid ) circumcenter, incenter, centroid ) circumeter, incenter, centroid ) orthocenter, centroid, circumcenter ) centroid, incenter, orthocenter 26) If, and are concurrent, with = 6, = 8, = 4, = 3, F = 2, and F = x, then the value of x is ) 1 ) 2 ) 3 ) 4 ) 5 F

7 27) If two angles of a triangle are 17 and 43, find the measure of the largest exterior angle. ) 60 ) 120 ) 163 ) 137 ) ) How many diagonals are in a convex polygon with nine sides? ) 9 ) 8 ) 7 ) 16 ) 27 29) Find the number of sides of a regular polygon if each exterior angle is 9. ) 9 ) 40 ) 36 ) 27 ) ) hords and intersect at a point inside a circle. If = 12, = 3, and is the midpoint of, then find the length of. ) 15 ) 36 ) 6 ) 12 ) 9 31) In the figure, is tangent to the circle at and intersects the circle again at. If is a point on arc remote from, m = 210, and m = 65, find m. ) 65 ) 145 ) 130 ) 80 ) ) If M is the midpoint of, M = 2x+3, and M = 3(x-2), find. ) 9 ) 18 ) 21 ) 42 ) 30

8 33) In the figure t is a transversal for parallel lines and m. If m 2 = x + y, m 5 = 3x+y, and x y = 15, find m ) 15 ) 20 ) 35 ) 45 ) l m t 34) In a regular tetrahedron planes parallel to each face pass through the midpoints of the remaining edges. If all such planes are considered at once, then how many smaller regular tetrahedra are formed? ) 2 ) 3 ) 4 ) 5 ) 6 35) If hexagon F has 60 degree rotational symmetry about its center P, then which one of the following is false? ) Quadrilateral P is a rhombus ) Quadrilateral F is an isosceles trapezoid ) Triangle has rotational symmetry of 60 ) The reflection of P about line gives FP ) Triangle P translated by vector P. 36) ircles with centers,,, and and common radius r are tangent to the circle with center. Find the circumference of the circle with center. ) ( ) r ) ) ) ) 4

9 37) Trapezoids RSNM and MNPQ are similar with RS = 3, m MRQ = m NSP = 90, MQ = NP = 5. Find the number of square units in the area of quadrilateral RSNM. ) 24 ) ) ) 12 ) 14 M N R S Q P 38) In square,, = 4, F = 9, and F = 8. Find the perimeter of the square. ) 15 ) 6 ) 23 ) 30 ) 21 F 39) Given: WXYZ is a trapezoid with, is the median, WX = 4x 7, MN = 2x + 10, and ZY = 2x + 1. Find the length of. ) 13 ) 18 ) 27 ) 36 ) 45 W X M N Z Y 40) In, bisects so that = 6, = 8, and = 4. Find the perimeter of. ) 14 ) 17 ) 18 ) 21 ) 28

10 xtras. 42 and 43 have been used in the test. 42) If the sides of a triangle are 20, 21, and 29 units, then the triangle is ) acute ) isosceles ) obtuse ) right ) not possible 43) Given: with =4, =6, =2, and =4. Find the area of. ) 14 ) 28 ) 56 ) 63 ) 12 44) If G is an isosceles trapezoid, G is a rectangle, F is the midpoint of, ==9, F=F=13, and G=10, find the area of G. ) 38 ) 44 ) 68 ) 120 ) 228