Download January Regional Geometry Team: Question #1 A regular n

January Regional
Geometry Team: Question #1
A regular n-gon has sides of length 2:
Let be the semi-perimeter of the polygon if n = 30.
Let be the perimeter of the polygon if n = 45.
Let be the length of the longest diagonal that can be drawn within a polygon when n = 8.
Let be the length of the apothem if n = 6.
Find
– .
January Regional
Geometry Team: Question #2
A trapezoid has a base of length 10, a midsegment of length 24 and legs of length 17 and 25.
Let be the length of the missing base.
Let be the perimeter of the trapezoid.
Let be the distance between the two bases.
Let be the tangent of the angle the leg of length 17 makes with the unknown base.
Find
–
–
.
January Regional
Geometry Team: Question #3
Find the sum of the hypotenuses of the 5 distinct right triangles with the smallest integral perimeters.
January Regional
Geometry Team: Question #4
In the figure to the right, AF║GR, ST is the midsegment
of Δ GRA and has a length of 7, SR has a length of 6
x° A w°
F
and GR has a length of 12.
Let be the length of line segment GA.
S
Let be x.
Let be w.
G
R
Find
.
T
137°
January Regional
Geometry Team: Question #5
Jackson loves folding paper. One day, he folds an 8” by 16” rectangular sheet of paper along the
diagonal to form a concave pentagon. What is the perimeter of the folded shape?
January Regional
Geometry Team: Question #6
. If a statement is true, add the corresponding value in parenthesis to . If the statement is
false, subtract the corresponding value in parenthesis from .
(-12) If the contrapositive statement is not true, the conditional statement is not true.
(3) Converses and Inverses are not the illogical nonequivalent.
(-7) A pair of lines can be both parallel and skew.
(9) The circumcenter is the point in a triangle where the angle bisectors meet.
(-4) The medians of a triangle separate the triangle into 6 triangles of equal area.
Find .
January Regional
Geometry Team: Question #7
A square is inscribed in a regular octagon with sides of length 4.
Let be the length of a side of the square.
Let be the length of the diagonal in the square.
Let be the perimeter of one of the trapezoids formed by
the space between the square and the octagon.
Find
.
January Regional
Geometry Team: Question #8
The golden ratio is the ratio of the length over the width in a rectangle in which the following
equation applies:
where is length and
is width. What is the golden ratio?
January Regional
Geometry Team: Question #9
A rectangle with sides α and β is inscribed in a semicircle with radius 6.
Let be the length of α if the ratio of α to β is 1.
β
Let be the length of α if the ratio of α to β is 2.
α
Let be the length of α if the ratio of α to β is ½.
α
Find
.
January Regional
Geometry Team: Question #10
In a 9-12-15 right triangle, an altitude is drawn to the hypotenuse. Within the larger of the two
triangles, another altitude is drawn to the hypotenuse. This process repeats to infinity. What is the
sum of all the altitudes?
January Regional
Geometry Team: Question #11
In Δ L1OL2,
is 90°, side L1L2 has length 23 and the value of L1 is
.
Let A be the value of the sine of L1.
Let B be the value of the sine of L2.
Find
.
January Regional
Geometry Team: Question #12
In a kite, two consecutive sides have lengths 29 and 35 and the diagonal joining those sides has
length 48. What is the length of the other diagonal?
January Regional
Geometry Team: Question #13
What is the sum of angles in a polygon with 345 sides?
January Regional
Geometry Team: Question #14
Given the coordinates of a triangle in 3-D, (23, 23, 23), (45, 45, 47) and (97, 55, -100):
Let be the x-coordinate of the centroid.
Let be the y-coordinate of the centroid.
Let be the z-coordinate of the centroid.
Find
.
January Regional
Geometry Team: Question #15
What is the area of a triangle with sides of lengths 21, 72 and 75?