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#1 Geometry – Hustle
FAMAT State Convention 2015
#3 Geometry – Hustle
FAMAT State Convention 2015
The altitude to the hypotenuse of a
30  60  90 triangle has length 6.
Find the length of the longer leg
of the triangle.
Find the area of a square with a diagonal of
length 9 2 .
Answer : _____________
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#2 Geometry – Hustle
FAMAT State Convention 2015
If A, B, and C are points on circle O,
AC is a diameter, and mAOB  60, find
the mACB.
Answer : _____________
Round
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#4 Geometry – Hustle
FAMAT State Convention 2015
Find the area of an isosceles trapezoid
with legs 5 and bases 9 and 17.
Answer : _____________
Answer : _____________
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#5 Geometry – Hustle
FAMAT State Convention 2015
#7 Geometry – Hustle
FAMAT State Convention 2015
A sphere has area 16 . Find its volume.
Find the volume of a right circular cone
that has a total surface area of 108 and
the diameter of the base is 12.
Answer : _____________
Answer : _____________
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#6 Geometry – Hustle
FAMAT State Convention 2015
#8 Geometry – Hustle
FAMAT State Convention 2015
A triangle has two sides of length
8 and 3. Find the sum of the possible
integer lengths of the third side that
would make the triangle obtuse.
The sides of a triangle are 10, 17, and 21. Find
the length of the altitude to the longest side.
Answer : _____________
Answer : _____________
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#9 Geometry – Hustle
FAMAT State Convention 2015
#11 Geometry – Hustle
FAMAT State Convention 2015
A rhombus has diagonals of length 2 5 and
4 5 . Find the length of the altitude of the
rhombus.
The sum of the interior angles of a convex
polygon is 6480  . Find the average measure
of an exterior angle of the polygon.
Answer : _____________
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#10 Geometry – Hustle
FAMAT State Convention 2015
Find the length of the radius of a circle
Answer : _____________
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#12 Geometry – Hustle
FAMAT State Convention 2015
with a central angle of 67  which
The legs of a right triangle are in the
ratio 1:2. If the area of the triangle
is 36, find the length of the hypotenuse.
Answer : _____________
Answer : _____________
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2
intercepts an arc of length 24 .
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#13 Geometry – Hustle
FAMAT State Convention 2015
#15 Geometry – Hustle
FAMAT State Convention 2015
The sum of the measures of the interior
angles of a convex polygon is 1620 .
Find the number of the diagonals of
the polygon.
A cone has a radius of 5 and slant height of 13.
A cylinder with radius 10 has the same volume
as the cone. Find the height of the cylinder.
Answer : _____________
Answer : _____________
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#14 Geometry – Hustle
FAMAT State Convention 2015
#16 Geometry – Hustle
FAMAT State Convention 2015
Find the average of the possible integer
measures of the third side of a triangle
with two sides having lengths 6 and 9.
Find the perimeter of isosceles FHJ with base
JH , K is the midpoint of JF and G is the
midpoint of HF , FK  2 x  3, GH  5x  9,
JH  4 x.
Answer : _____________
Answer : _____________
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#17 Geometry – Hustle
FAMAT State Convention 2015
#19 Geometry – Hustle
FAMAT State Convention 2015
A right hexagonal prism has a volume
of 96 3 . All 18 edges are congruent.
Find the total surface area of the prism.
Find the area of an equilateral triangle
inscribed in a circle of radius 1.
Answer : _____________
Answer : _____________
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#18 Geometry – Hustle
FAMAT State Convention 2015
#20 Geometry – Hustle
FAMAT State Convention 2015
A right triangle with legs 3 and 2 3
is revolved about its longer leg.
Find the volume of the solid generated
by this rotation.
Six times the volume of a cube is equal
to the sum of its total surface area and the
total length of its edges. Find the length
of one edge.
Answer : _____________
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Answer : _____________
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#21 Geometry – Hustle
FAMAT State Convention 2015
#23 Geometry – Hustle
FAMAT State Convention 2015
A chord with length 24 has an arc of 120 .
Find the distance of the chord from the center
of the circle.
Find the length of the longest straight metal rod
that can be placed in a rectangular box with
dimensions 6, 8, and 10.
Answer : _____________
Answer : _____________
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#22 Geometry – Hustle
FAMAT State Convention 2015
#24 Geometry – Hustle
FAMAT State Convention 2015
Two concentric circles have radii of
7 and 13. Find the length of a chord
of the larger circle that is tangent to
the smaller circle.
A convex octagon has angles of measure
100 , 80, 150 and 130 . If the remaining
angles are in the ratio 14:15:16:17, find the
measure of the largest angle of the octagon.
Answer : _____________
Answer : _____________
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#25 Geometry – Hustle
FAMAT State Convention 2015
Given right triangle ABC with hypotenuse AC .
D is a point on AC such that BD  AC .
AC  16, BA  8 3 . Find the value of BD  BC .
Answer : _____________
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Solutions for Geometry Hustle
FAMAT State 2015
1. 12
2. 30 or 30
3. 81
4. 39
5.
32

3
6. 32
7. 72 3 or 72 3
8. 8
9. 4
10. 64
180
9
11.
or 9
19
19
12. 6 5
13. 44
14. 9
15. 1
16. 60
17. 96 48 3
18. 6 3 or 6 3
3 3
3
3 or
19.
or
4
4
0.75 3
20. 2
21. 4 3
22. 4 30
23. 10 2
24. 170
25. 4 3  8 or 8  4 3
#1
Let’s name this ABC with the right
angle at C, 30  angle at A. The altitude intersects
the hypotenuse at point E. In ACE , CE = 6, AC
= 12, which is the longest leg of the triangle.
#2
The measure of arc AB is 60  , making
mACB  30 .
#3
Side is 9, area is 81.
#4
Draw the altitudes of the trapezoid and
use the right triangles determined by the
altitudes. The hypotenuse is 5. Since the short
base is 9, the lower base has the legs of the right
triangles as 4. Using a triangle with hyp 5 and
leg 4, makes the other leg, which is the altitude,
have a length of 3. So, the area is
1
 3   9  17   39.
2
#5
4 r 2  16 , r 2  4, r  2, 2 .
4
32
V    23   .
3
3
#6
The third side of the triangle is between 5
and 11. So, we need to check each of those
possible sides with the other two with lengths of
reminder that the square of the largest side is
greater than the sum of the squares of the other
two sides.
6,8,3 64 > 36+9 so
6 works.
7,8,3 64>49+9 so
7 works.
8,8,3
64  64+9
8 does not work.
9,8,3 81>64+9
9 works.
10,8,3 100>64+9
10 works.
The sum of 6+7+9+10 is 32.
#7
TSA=  r 2   rl  180 ; canceling  and
moving substituting 6 for the radius makes the
slant height equal 12, and the altitude 6 3 .
Now to find the volume:
1
1
Bh   36  6 3  72 3 .
3
3
#8
Using Heron’s formula find the
semiperimeter of the triangle to be 24.
1
24  14  7  3   21  7, h  8 .
2
4  5  6  7  8  9  10  11  12  13  14
9.
11
Solutions for Geometry Hustle
FAMAT State 2015
#9
Find the area of the rhombus using the
diagonal formula, then work backwards from
1
1
A=bh. d1d 2   2 5  4 5  20 . Now find
2
2
the length of a side of the rhombus using
Pythagorean Theorem and one of the 4 triangles
formed by drawing the diagonals.
2 5    5 
2
2
 20  5  25. Square root of 25
is 5. 20  5h, h  4.
#10
The fractional part of the circle
multiplied by the circumference gives the arc
length. So work this backwards.
1
67
2  2 r  24 , r  64. .
360
#11
Find the number of sides for the polygon
and divide 360 (sum of exterior angles of all
polygons) to find the average degree measure.
360 180

6480   n  2180, n  38.
38 19.
#12
Since the sides are in the ratio of 1:2, let
1
the sides be x and 2x .  x  2 x  36, x  6. The
2
sides of the triangle are 6 and 12 . Use
Pythagorean Theorem to find the hypotenuse is
6 5.
#13
Find the number of sides:
1620   n  2 180, n  11. Then number of
diagonals
11  8
 44.
2
#14
The 3rd side must be between 3 and 15.
The average of the possible integers
#15
The slant height of the cone is 13 and the
radius is 5 so the altitude is 12. Volume of the
1
cone is  25  12  100 .
3
Volume of the cylinder is 100  h . Set these
two volumes equal to find the height of the
cylinder is 1.
#16
Since JF  FH , and K and G are
midpoints of those two sides, the 4 segments are
equal.
2 x  3  5x  9, x  4.
JF  FH  22, JH  16,44  16  60.
#17
Find the length of one edge.
6e 2 3
 e, e  4. Now find the lateral
4
surface area by multiplying the perimeter of the
base with the height: 24  4  96 . The area of the
96 3 
two bases is 2  6 
s2 3
16 3
 12 
 48 3 ,
4
4
making the total surface area 96  48 3 .
#18
When rotated, a cone is formed with a
height of 2 3 and radius of the base of 3.
1
 9  2 3  6 3 .
3
#19
Draw two radii to two vertices of the
triangle. This forms a triangle with angles of 3030-120. Draw the altitude of this triangle to find
1
its length is . Using one of those right
2
1
3 and the sides of
triangles, the other leg is
2
the original triangle is
3 . Using
s2 3 3 3

.
4
4
#20
6s 3  6s 2  12s,6s  s 2  s  2   0;
6s  s  2  s  1  0; s  0 ,2, 1 .
Solutions for Geometry Hustle
FAMAT State 2015
#21
Draw the chord and the segment that is
the distance from the center. The central angle is
120  and when the segment from the center to
the chord is drawn, it forms two 30-60-90
triangles with the side opposite the 60  angle
having a length of 12. Divide this by 3 to get
the length of the segment from the center to the
chord.
12
3 12 3


 4 3.
3
3
3
#22
Draw the radius of the smaller circle to
the point of tangency forming a right angle.
Draw the radius of the larger circle from the
center to the endpoint of the tangent chord. This
gives a right triangle with hypotenuse 13 and leg
7. Use Pythagorean theorem to find the other leg.
x 2  49  169, x 2  120, x  2 30 . Double this to
find the length of the chord 4 30 .
#23
l 2  w2  h2 ; 36  64  100  200  10 2 .
#24
Find the sum of the angles of an octagon:
 6180  1080. Subtract the sum of the angles
that are known:
1080  100  80  150  130  620 . Now use the
remaining ratios:
14 x  15x  16 x  17 x  620, x  10. Angle
measures are 100,80,150,130,140,150,160,170.
Largest is 170.
#25
After drawing the diagram, since 8 3 is
1
3 times 16, these triangles are 30-60-90
2
triangle with A  30, C  60 and the measures
of the angles of the smaller triangles can be
found. Since BC is opposite the 30  angle, its
length is 8. The length of BD is also needed. In
ABD , AB is the hypotenuse with a length of
8 3 and BD is opposite the 30 angle so it is
half of that measure 4 3 . BD  BC  4 3  8 .