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February Invitational Geometry Team Questions February Invitational Geometry Team Questions FEBRUARY STATEWIDE GEOMETRY TEAM #1 Identify the following statements as true or false. Find the number of false statements plus 10 times the number of true statements. Two planes always intersect in a line. Two coplanar lines that do not intersect are parallel. The slope of perpendicular lines are reciprocals. All triangles have exactly 2 acute angles. FEBRUARY STATEWIDE GEOMETRY TEAM #1 Identify the following statements as true or false. Find the number of false statements plus 10 times the number of true statements. Two planes always intersect in a line. Two coplanar lines that do not intersect are parallel. The slope of perpendicular lines are reciprocals. All triangles have exactly 2 acute angles. FEBRUARY STATEWIDE GEOMETRY TEAM #2 The number of diagonals in an undecagon The sum of the interior angles of an undecagon The maximum number of distinct regions that can be formed by 11 lines in a plane, given that no lines are parallel and only two lines intersect at a single point. Find B – C – A. FEBRUARY STATEWIDE GEOMETRY TEAM #2 The number of diagonals in an undecagon The sum of the interior angles of an undecagon The maximum number of distinct regions that can be formed by 11 lines in a plane, given that no lines are parallel and only two lines intersect at a single point. Find B – C – A. FEBRUARY STATEWIDE GEOMETRY TEAM #3 In right ABC , with right angle A, point D lies on segment AB and point E lies on BC such that ABC DBE . If DB 5 , AC 10 , and DE 6 , find DA . FEBRUARY STATEWIDE GEOMETRY TEAM #3 In right ABC , with right angle A, point D lies on segment AB and point E lies on BC such that ABC DBE . If DB 5 , AC 10 , and DE 6 , find DA . FEBRUARY STATEWIDE GEOMETRY TEAM #4 A square is inscribed in a semicircle such that one side of the square lies on the diameter of the semicircle. The radius of the semicircle is 30 units. Find the area of the square Find the perimeter of the semicircle Find the area inside the semicircle, but outside the square Find A + B + C. FEBRUARY STATEWIDE GEOMETRY TEAM #4 A square is inscribed in a semicircle such that one side of the square lies on the diameter of the semicircle. The radius of the semicircle is 30 units. Find the area of the square Find the perimeter of the semicircle Find the area inside the semicircle, but outside the square Find A + B + C. FEBRUARY STATEWIDE GEOMETRY TEAM #5 A 4 unit cube is painted blue, then cut into smaller cubes with 1 unit sides. How many of the smaller cubes have exactly 2 sides painted blue? How many of the smaller cubes have exactly 1 side painted blue? How many of the smaller cubes have no sides painted blue? Find A C . B FEBRUARY STATEWIDE GEOMETRY TEAM #5 A 4 unit cube is painted blue, then cut into smaller cubes with 1 unit sides. How many of the smaller cubes have exactly 2 sides painted blue? How many of the smaller cubes have exactly 1 side painted blue? How many of the smaller cubes have no sides painted blue? Find A C . B FEBRUARY STATEWIDE GEOMETRY TEAM #6 Give the terms corresponding to the following definitions respectively. Intersection of the medians of a triangle Intersection of the angle bisectors of a triangle Intersection of the altitudes of a triangle Intersection of the perpendicular bisectors FEBRUARY STATEWIDE GEOMETRY TEAM #6 Give the terms corresponding to the following definitions respectively. Intersection of the medians of a triangle Intersection of the angle bisectors of a triangle Intersection of the altitudes of a triangle Intersection of the perpendicular bisectors FEBRUARY STATEWIDE GEOMETRY TEAM #7 Mr. Kite wants to measure the length of the chain on his bicycle. He knows that the gear on his pedals is 10 cm., and the gear on the rear wheel is 4 cm. and the distance between the center of the gears is 30 cm. Find the exact length of the chain. FEBRUARY STATEWIDE GEOMETRY TEAM #7 Mr. Kite wants to measure the length of the chain on his bicycle. He knows that the gear on his pedals is 10 cm., and the gear on the rear wheel is 4 cm. and the distance between the center of the gears is 30 cm. Find the exact length of the chain. FEBRUARY STATEWIDE GEOMETRY TEAM #8 How many of the following statements prove that a quadrilateral MUST be a parallelogram? Opposite sides are parallel Diagonals are congruent Opposite angles are congruent One pair of sides that are both parallel and congruent Two acute angles and two obtuse angles A diagonal creates two congruent triangles FEBRUARY STATEWIDE GEOMETRY TEAM #8 How many of the following statements prove that a quadrilateral MUST be a parallelogram? Opposite sides are parallel Diagonals are congruent Opposite angles are congruent One pair of sides that are both parallel and congruent Two acute angles and two obtuse angles A diagonal creates two congruent triangles FEBRUARY STATEWIDE GEOMETRY TEAM #9 How many of the following constructions can be performed using only an unmarked straightedge and a compass? a. b. c. d. e. f. g. h. i. the bisector of an angle given an angle, its bisector the trisectors of an angle given an angle, two rays that trisect it the midpoint of a segment two trisectors of a segment the perpendicular bisector of a segment the center of a circle a 30-60-90 triangle a regular quadrilateral a regular pentagon FEBRUARY STATEWIDE GEOMETRY TEAM #9 How many of the following constructions can be performed using only an unmarked straightedge and a compass? a. b. c. d. e. f. g. h. i. the bisector of an angle given an angle, its bisector the trisectors of an angle given an angle, two rays that trisect it the midpoint of a segment two trisectors of a segment the perpendicular bisector of a segment the center of a circle a 30-60-90 triangle a regular quadrilateral a regular pentagon FEBRUARY STATEWIDE GEOMETRY TEAM #10 A = the measure of the angle formed by two secant lines that intersect two arcs in a circle of measure 30° and D°. B = the measure of the angle inscribed in a semicircle of radius A cm. C = the measure of the angle formed by two tangent lines that intersect an arc in a circle of B measure . 3 D = the measure of the central angle that intersects the same arc in a circle as an inscribed angle of measure C° Determine A + B + C + D. FEBRUARY STATEWIDE GEOMETRY TEAM #10 A = the measure of the angle formed by two secant lines that intersect two arcs in a circle of measure 30° and D°. B = the measure of the angle inscribed in a semicircle of radius A cm. C = the measure of the angle formed by two tangent lines that intersect an arc in a circle of B measure . 3 D = the measure of the central angle that intersects the same arc in a circle as an inscribed angle of measure C° Determine A + B + C + D. FEBRUARY STATEWIDE GEOMETRY TEAM #11 A = the area of a rhombus with diagonals 5 units and 11 units B = the area of a segment determined by a 120° central angle in a circle with circumference, 20π units C = the area of a regular hexagon with apothem 4 units D = the area of a scalene triangle formed by two vertices of a square and a third point lying 8/9 the distance between the other two vertices. The square has perimeter 8 units. List the letters, A-D, in order from least to greatest area. FEBRUARY STATEWIDE GEOMETRY TEAM #11 A = the area of a rhombus with diagonals 5 units and 11 units B = the area of a segment determined by a 120° central angle in a circle with circumference, 20π units C = the area of a regular hexagon with apothem 4 units D = the area of a scalene triangle formed by two vertices of a square and a third point lying 8/9 the distance between the other two vertices. The square has perimeter 8 units. List the letters, A-D, in order from least to greatest area. FEBRUARY STATEWIDE GEOMETRY TEAM #12 A = the lateral area of a right cone with base radius of 3 units and height of 4 units B = the volume of an oblique cylinder with height of 4 units and bases with perimeter 7π units C = the surface area of a tetrahedron with edge length 6 units D = the volume of a spherical cap formed by planar cross section 1 unit away from the center of a sphere with radius 9 units Determine A + B + C + D. FEBRUARY STATEWIDE GEOMETRY TEAM #12 A = the lateral area of a right cone with base radius of 3 units and height of 4 units B = the volume of an oblique cylinder with height of 4 units and bases with perimeter 7π units C = the surface area of a tetrahedron with edge length 6 units D = the volume of a spherical cap formed by planar cross section 1 unit away from the center of a sphere with radius 9 units Determine A + B + C + D. FEBRUARY STATEWIDE GEOMETRY TEAM #13 What is the area of the circle inscribed in a triangle with sides 9, 40, 41? FEBRUARY STATEWIDE GEOMETRY TEAM #13 What is the area of the circle inscribed in a triangle with sides 9, 40, 41? FEBRUARY STATEWIDE If GEOMETRY TEAM #14 is the number of correct statements below, find The converse of The inverse of is is The contrapositive of is The contrapositive of is FEBRUARY STATEWIDE If GEOMETRY TEAM #14 is the number of correct statements below, find The converse of The inverse of The contrapositive of The contrapositive of is is is is