February Regional
Geometry Individual Test
The acronym NOTA is defined as “none of the above answers is correct”. Good Luck and Have fun! Note
that the diagrams are not drawn to scale. Remember everything in geometry can be broken down into
simple constructions and patterns.
1.
Joy is standing 400 ft from the base of a tower, looking up towards a Multivariable Calculus
book situated at the top of the tower. The angle of elevation from his line of sight is measured to
be 60 . Assuming Joy’s line of sight is 4 12 feet, from the ground, find the height in feet of
the tower.
A.
B.
E. NOTA
D.
C.
2. A regular octagon is drawn with a side length of 8 . Find its area.
A.
B.
E. NOTA
C.
D.
3. A triangle is drawn with side lengths 24, 7, and 25. Find the product of the largest and smallest
altitude.
A.
B.
C.
D.
E. NOTA
4. Andres is creating plans to construct a boardwalk around a park. He draws a rectangle to
represent the park with a length of 6 units and a width of 8 units. If the boardwalk has a uniform
width of 5 units, find the area enclosed by the boardwalk.
A.
5.
B.
C.
D.
E. NOTA
Let AB  6 , BH  4 , CB  16 , DH  8 , BE  4 , CD  3 , AH  m , and DE  n .
Evaluate m  n .
A
B
H
E
D
C
A. 1
B. 2
C. 4
D. 3
E. NOTA
6. From a point C , Connor is moving west at 5km per hour and Nick is heading north at
2km per hour . If Nick began moving from point C at noon and Connor began moving at
1: 00 pm , find the shortest distance between the two at 4 : 00 pm .
A.
B.
C. 17
D.
E. NOTA
February Regional
Geometry Individual Test
For numbers 7-10 consider a triangle,
7.
XYZ , that satisfies the relation XYZ  1.5YZX  3ZXY
Find the value of the angles XYZ , ZXY , YZX respectively
A. 90,60,30
8. Find the area of
A. 54
B. 90,30,60
C. 30,60,90
D. 30,90,60
E. NOTA
C. 27
D.
E. NOTA
XYZ if XZ  6 .
B.
9. Let p represent the length of the altitude drawn from point Y to XZ . Find p given the area of
area of XYZ  6 3 units 2 .
A.
C.
B.
E. NOTA
D.
10. Find the ratio represented by the length of XZ over the length of YX .
B.
A.
C.
E. NOTA
D.
11. Three circles, each of radius 6s , are drawn externally tangent to each other. A band is wrapped
tightly around the outside of all the circles. Find the length of the band in terms of s .
E. NOTA
C.
D.
A.
B.
12. Two circles of radii 6 and 10 are drawn externally tangent to each other. Let x represent the
length of the common external tangent. Find 2x .
A.
B.
13. In
C.
E. NOTA
D.
S , BW  12 , DR  30 , BD  20 ,
BW  DR , and DR is the diameter. Find the
length of RW .
D
S W
R
B
A. 14
B. 16
C.
D. 4
E. NOTA
D. 13
E. NOTA
14. Find the distance between the points  5, 2  and 10,10  .
A. 17
B. 5
C.
February Regional
Geometry Individual Test
15. John circumscribes a circle about an equilateral triangle of side length s . He then circumscribes
a square about the circle. Let b represent the area bounded between the square and the circle, and
let a represent the area bounded between the circle and the triangle. Find a  b .
 16  3 3 
 12 


C. s 2 
 16  3 3 
 12 


 43 3 
 12 


D. s 2 
A. s 2 
E. NOTA
 8  3 3  16 


12


B. s 2 
For questions 16-18 consider a triangle, ABC . Let the semiperimeter of ABC  16 and the side
lengths be written as x  5 , x  1 , and x  3 .
16. Find the value of x
A.
B. 14
23
3
D. 13
41
3
C.
E. NOTA
17. Find the area of ABC
A.
18. Classify
B.
C.
D.
E. NOTA
B. Right
C. Isosceles
D. Acute
E. NOTA
ABC .
A. Obtuse
19. Lirun draws three circles of radius r in an overlapping manner such that the distance between
their centers is r . He then finds the area of the intersection within all three circles, denoted as R
in the diagram, in terms of r . Assuming he solved for the area correctly, what is Lirun’s answer?
R

r 2 2  5 3
A.

4
r  2 3
2
B.
4



r 2 4  3 3
C.

12
r  3
2
D.
2


E. NOTA
February Regional
Geometry Individual Test
20. A circle can sometimes be inscribed within which of the following?
I.
II.
Rectangle
Rhombus
III.
IV.
A. II, IV, and V
B. I, II, IV, and V
Kite
Parallelogram
V.
C. I, II, III, IV, and V
D. I, II, III, and V
Trapezoid
E. NOTA
21. Emerald is trying to balance a triangular pan on top of Adrianne’s head. She is having difficulty
balancing the pan so Brelbi helps her out by telling her to balance the pan at it center of gravity.
In Euclidean Geometry this point is referred to as the
?
A. Nine Point Center
B. Centroid
C. Orthocenter
D. Circumcenter
E. NOTA
22. Find the converse of the inverse of the contrapositive of the statement “If Ivan studies PreCalculus like a scholar, then Ivan will rank in Pre-Calculus”
A.
B.
C.
D.
E.
If Ivan does not study Pre-Calculus like a scholar, then Ivan will not rank in Pre-Calculus
If Ivan does not rank in Pre-Calculus, then Ivan did not study Pre-Calculus like a scholar
If Ivan ranks in Pre-Calculus, then Ivan studied Pre-Calculus like a scholar
If Ivan studies Pre-Calculus like a scholar, then Ivan will rank in Pre-Calculus
NOTA
23. Find the length of b in terms of x and h .
c
b
h
B.
A.
C.
D.
a
x
E. NOTA
24. Find the length of a chord located 6 units away the center of a circle with a radius of 10 units
A. 8
B. 16
C.
D.
E. NOTA
D. 27
E. NOTA
25. How many diagonals are there in a heptagon?
A. 21
B. 14
C. 28
26. Two concentric circles are drawn such that the difference between the radius of the larger circle
and smaller circle is 5. Let x represent the radius of the smaller circle and y represent the radius
of the larger circle. Find the area of the annulus formed if x  y  9 and it is given that x and y
are integral values
A.
B.
C.
D.
E. NOTA
February Regional
Geometry Individual Test
27. Kelly and Sara investigate the applications of  using a simple geometric
C
D
construction involving a square inscribed a semicircle. The square,
CDRS , is drawn such that RS coincides with the diameter of
RS
the semicircle. Find the value of phi (  ) by finding
.
BR
A. 1  5
B.
1 5
4
For questions 28-30 consider a square,

2
B.
 2
16
1 2
   2 
2
B.
1
   2 
2
C.
C.
 2
2
2
B.
2
D.
P and
E. NOTA
P
.
C. 2 2

E. NOTA
2
ABCD . Find h in terms of  .
1 2
   2 
2
30. Find the circumference of the circle in terms of
A.
1 5
2
D.
R
.
29. Let h represent the area formed between
A.
S
ABCD , with a side length of  inscribed in
28. Find the area of the circle in terms of
A. 2
3 5
2
C.
A
D.
1 2
   4 
2
D.
2
E. NOTA
E. NOTA
B