Download Test - FloridaMAO

January Regional
Geometry Team: Question #1
1. How many of the following statements are true (in Euclidean geometry)?
I. Any three points in a plane are always collinear.
II. Two coplanar lines that are not parallel must intersect.
III. Given a line  and a point P not on  , there is exactly one line through P that
is parallel to  .
IV. Given a line  and a point P not on  , there is exactly one line through P that
is perpendicular to  .
V. Two lines in space can intersect at two distinct points.
January Regional
Geometry Team: Question #1
1. How many of the following statements are true (in Euclidean geometry)?
I. Any three points in a plane are always collinear.
II. Two coplanar lines that are not parallel must intersect.
III. Given a line  and a point P not on  , there is exactly one line through P that
is parallel to  .
IV. Given a line  and a point P not on  , there is exactly one line through P that
is perpendicular to  .
V. Two lines in space can intersect at two distinct points.
January Regional
Geometry Team: Question #2
2. For each of the following parts, two angles of a triangle are given. The answer to each
part is the measure of the other angle of the triangle. If a triangle with the given two
angles cannot exist, the answer to that part is 180.
A.
B.
C.
D.
35 and 40
83 and 97
69 and 101
78 and 98
Find the sum of the answers to parts A, B, C, and D.
January Regional
Geometry Team: Question #2
2. For each of the following parts, two angles of a triangle are given. The answer to each
part is the measure of the other angle of the triangle. If a triangle with the given two
angles cannot exist, the answer to that part is 180.
A.
B.
C.
D.
35 and 40
83 and 97
69 and 101
78 and 98
Find the sum of the answers to parts A, B, C, and D.
January Regional
Geometry Team: Question #3
3. In triangle ABC, B  90 .
W = the length of AC if AB  9 and BC  12
X = the length of AB if BC  12 and AC  13
Y = the length of AC if AB  3 2 and C  45
Z = the length of AB if AC  8 and C  30
Find W  X  Y  Z .
January Regional
Geometry Team: Question #3
3. In triangle ABC, B  90 .
W = the length of AC if AB  9 and BC  12
X = the length of AB if BC  12 and AC  13
Y = the length of AC if AB  3 2 and C  45
Z = the length of AB if AC  8 and C  30
Find W  X  Y  Z .
January Regional
Geometry Team: Question #4
4. A standard 12-hour analog clock currently reads 12:00. After 2009 hours, what time
will it read? Ignore a.m. and p.m.
January Regional
Geometry Team: Question #4
4. A standard 12-hour analog clock currently reads 12:00. After 2009 hours, what time
will it read? Ignore a.m. and p.m.
January Regional
Geometry Team: Question #5
5. A great diagonal of a polygon is a diagonal that divides the polygon into two
polygons of equal area. For positive integers n, n  3 , let g (n) denote the number of
great diagonals of a regular n-gon. What is g (19)  g (20) ?
January Regional
Geometry Team: Question #5
5. A great diagonal of a polygon is a diagonal that divides the polygon into two
polygons of equal area. For positive integers n, n  3 , let g (n) denote the number of
great diagonals of a regular n-gon. What is g (19)  g (20) ?
January Regional
Geometry Team: Question #6
height
6. Albert has a rectangular piece of paper whose height is shorter than its width, as
shown. He folds the paper in half along a fold parallel to the height of the paper. The
resulting rectangle is similar to the original piece of paper.
width
What is the width of the original paper divided by its height?
January Regional
Geometry Team: Question #6
height
6. Albert has a rectangular piece of paper whose height is shorter than its width, as
shown. He folds the paper in half along a fold parallel to the height of the paper. The
resulting rectangle is similar to the original piece of paper.
width
What is the width of the original paper divided by its height?
January Regional
7.
Geometry Team: Question #7
P = the height of isosceles trapezoid ABCD if AB || CD , AB  10 , CD  16 , and
ADC  45
Q = the area of parallelogram ABCD if AB  6 and the perpendicular from point
B to line CD has length 4
R = the length of the shorter diagonal of rhombus ABCD if AB  4 and
ABC  60
S = the length of a diagonal of a rectangle with side lengths 8 and 15
Find P  Q  R  S .
January Regional
7.
Geometry Team: Question #7
P = the height of isosceles trapezoid ABCD if AB || CD , AB  10 , CD  16 , and
ADC  45
Q = the area of parallelogram ABCD if AB  6 and the perpendicular from point
B to line CD has length 4
R = the length of the shorter diagonal of rhombus ABCD if AB  4 and
ABC  60
S = the length of a diagonal of a rectangle with side lengths 8 and 15
Find P  Q  R  S .
January Regional
Geometry Team: Question #8
8. If a triangle has side lengths a, b, and c, then the length of the median to the side of
length c is given by
2a 2  2b2  c 2
.
2
The sum of the squares of the median lengths of a triangle is 3. What is the sum of the
squares of the side lengths of the triangle?
January Regional
Geometry Team: Question #8
8. If a triangle has side lengths a, b, and c, then the length of the median to the side of
length c is given by
2a 2  2b2  c 2
.
2
The sum of the squares of the median lengths of a triangle is 3. What is the sum of the
squares of the side lengths of the triangle?
January Regional
Geometry Team: Question #9
9. A rectangle has integer side lengths and area 12. What is the sum of all possible
diagonal lengths of the rectangle?
January Regional
Geometry Team: Question #9
9. A rectangle has integer side lengths and area 12. What is the sum of all possible
diagonal lengths of the rectangle?
January Regional
10. Suppose P is a regular 17-gon.
I = the measure of an interior angle of P
J = the sum of the interior angles of P
K = the measure of an exterior angle of P
L = the sum of the exterior angles of P
Find ( I  J  K  L) 180 .
Geometry Team: Question #10
January Regional
10. Suppose P is a regular 17-gon.
I = the measure of an interior angle of P
J = the sum of the interior angles of P
K = the measure of an exterior angle of P
L = the sum of the exterior angles of P
Find ( I  J  K  L) 180 .
Geometry Team: Question #10
January Regional
Geometry Team: Question #11
11. ABCD is a convex quadrilateral. The perpendicular bisectors of sides AB and BC
intersect at E, and the perpendicular bisectors of sides CD and AD intersect at F, so that E
and F are on opposite sides of AC. AC  24 , AE  13 and AF  15 . What is EF?
January Regional
Geometry Team: Question #11
11. ABCD is a convex quadrilateral. The perpendicular bisectors of sides AB and BC
intersect at E, and the perpendicular bisectors of sides CD and AD intersect at F, so that E
and F are on opposite sides of AC. AC  24 , AE  13 and AF  15 . What is EF?
January Regional
12. Two rays R1 and R2 have the same
endpoint A0. The angle between the rays is 1.
Point A1 is on R1, point A2 is on R2, point A3 is
on R1, point A4 is on R2, and so on up to point
An, so that A0, A1, A2, … , An are all different
and A0 A1  A1 A2  A2 A3    An1 An . What is
the largest possible value of n?
Geometry Team: Question #12
A0
A1
A2
A3
A4
A5
R1
R2
January Regional
12. Two rays R1 and R2 have the same
endpoint A0. The angle between the rays is 1.
Point A1 is on R1, point A2 is on R2, point A3 is
on R1, point A4 is on R2, and so on up to point
An, so that A0, A1, A2, … , An are all different
and A0 A1  A1 A2  A2 A3    An1 An . What is
the largest possible value of n?
Geometry Team: Question #12
A0
A1
A2
A3
A4
A5
R1
R2
January Regional
Geometry Team: Question #13
13.
M = the minimum number of regions 2009 lines in a plane divide the plane into
N = the maximum number of regions 3 circles in a plane divide the plane into
Find M  N .
January Regional
Geometry Team: Question #13
13.
M = the minimum number of regions 2009 lines in a plane divide the plane into
N = the maximum number of regions 3 circles in a plane divide the plane into
Find M  N .
January Regional
14.
A  sin 30  cos 30
B  sin 45  cos 45
C  sin 60  cos 60
Find A  B  C
Geometry Team: Question #14
January Regional
14.
A  sin 30  cos 30
B  sin 45  cos 45
C  sin 60  cos 60
Find A  B  C
Geometry Team: Question #14
January Regional
Geometry Team: Question #15
15. ABC is a right triangle with A  90 . Point D is the foot of the altitude drawn from
A to hypotenuse BC. How many of the following statements are true?
I. Triangles ADB and CDA are similar.
II. Triangles ADB and CAB are similar.
III. Triangles ADC and BAC are similar.
January Regional
Geometry Team: Question #15
15. ABC is a right triangle with A  90 . Point D is the foot of the altitude drawn from
A to hypotenuse BC. How many of the following statements are true?
I. Triangles ADB and CDA are similar.
II. Triangles ADB and CAB are similar.
III. Triangles ADC and BAC are similar.