Theta Circles
FAMAT State Convention 2011
Solutions
1.
.
2.
.
Measure of arc
3.
4.
5.
If an equiangular polygon inscribed in a circle has an odd number of sides, it is also equilateral.
If a line is perpendicular to a plane of a circle and passes through the circle’s center, any point on the
line is equidistant from any two points of the circle. Triangle
is a
triangle with
. Using the side ratios, the radius of the circles is 4. Therefore the area is
.
The radius is the distance from the center to the line.
Therefore the area of the circle is
6.
.
.
For every chord between two points there are two arcs, so
7.
8.
.
arcs.
.
Theta Circles
FAMAT State Convention 2011
Solutions
9.
The resulting solid is a cylinder with an inner cylinder removed.
.
10.
11.
12.
.
.
Draw a segment from a vertex of the triangle to the center of the closest circle. Draw another
segment from that circle perpendicular to a side. This forms a
triangle. You will find
the length from the vertex to the point along the side of the triangle to the segment perpendicular to
the center to be
. Repeat this process to find that a side length of the triangle is
.
There the Perimeter is
.
13.
I – Always, II – Sometimes, III – Sometimes, IV – Sometimes.
14.
For each point
chords can be drawn. Since there are 10 points, there are
chords. But each chord has been included twice, so
15.
16.
chords.
There exists a chord of the circle connecting the x-intercepts. A line segment can be drawn from the
center perpendicular to this chord and will bisect it. Bisecting this chord give you the x-coordinate, 25,
this is also the radius of the circle. Drawing a triangle with a radius as the hypotenuse, base as half of
the chord, and height representing the y-coordinate, you find the points. Two points work with the
given information:
.
Any chord that has one endpoint at the point of tangency is bisected by the smaller circle.
17.
. There are two ways to find
the area of a triangle formed by the diagonals of the rhombus. By setting the areas equal, you can solve
for the radius of the circle.
. Notice that the radius of the circle is also
the height of one of the triangles.
18.
. Back substituting, you will find the measure of the third arc to be
.
Theta Circles
FAMAT State Convention 2011
Solutions
19.
20.
Using the distance formula, the distance between the centers is 13. Filling in the rest of the lengths,
you have a 5-12-13 triangle. Therefore, the length of the common internal tangent is 12.
Let x be the radius of the inscribed circle. The hypotenuse is the diameter of the circumscribed circle.
Thus you have the equation
.
.
21.
If you draw a segment from the corner to the center of the can, you will have the diagonal of a
square. The length of the rope is the diagonal – radius. Therefore,
22.
23.
and
with
. Therefore,
.
Using the Law of Cosines,
.
.
24.
If you connect the centers of the smaller circles, you have a square with side 10. The diagonal of the
square is
. This means the radius of the larger circle is
.
.
25.
26.
.
Using Brahmagupta’s formula,
27.
.
,
.
28.
29.
30.
.
Using Ptolemy’s Theorem,
.