Download February Invitational Team Questions February

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