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January Regional
Geometry Individual Test
The abbreviation NOTA denotes “none of these answers”. All numbers are assumed real. Diagrams may not
necessarily be drawn to scale. Good luck!
1. Given a regular polygon with 23 sides, how many diagonals does the polygon have?
A. 230
B. 253
C. 276
D. 299
E. NOTA
2. What is the measure of an angle in degrees such that 3 times its compliment is equal to 3/5ths of its
supplement minus ½ of itself?
A. 1620/19
B. 1296/17
C. 900/11
D. 600/17
E. NOTA
3. Given the 3-4-5 triangle Δ ABC to the right, find the length of n,
B
the altitude drawn from point D, the intersection of the medians drawn
from vertices A and B, down to AC.
A. 1
3
B. √3/2
C. √13/3
D. 2
A
E. NOTA
4. Which of the following will always be in the interior of the triangle?
I) Incenter
II) Circumcenter
III) Orthocenter
IV) Centroid
A. IV only
B. I & IV
C. II & III
5
D
n
C
4
D. I, III & IV
E. NOTA
5. George, who lives in a gridded world, is hungry and wants to go get cupcakes and frozen yogurt. If he is
currently at the point (0, 5) and the cupcake store is at the point (12, 7) and there is a frozen yogurt store at
every location along the line y=0, what is the shortest distance he can travel to the cupcake store if he gets
frozen yogurt along the way?
A. 10
B. 2√37
C. 12√2
D. 13
E. NOTA
6. Given the parallelogram to the right, what is the measure of the angle
labeled 2y (all measures are presented in degrees)?
A. 45
B. 55
C. 57.5
D. 60
E. NOTA
4x+2y
2y
4x
z
7. The shortest leg of a 30-60 right triangle measures 2 ft. What is the length of the hypotenuse in inches?
A. 2√3
B. 4
C. 24√3
D. 48
E. NOTA
8. Given the statement, “if your name is Steve, you like drawing giraffes with obnoxiously long necks,” what
is the converse of the contrapositive of the inverse of the converse of the contrapositive of the statement?
A. If your name is Steve, you like drawing giraffes with obnoxiously long necks.
B. If your name is not Steve, you do not like drawing giraffes with obnoxiously long necks.
C. If you like drawing giraffes with obnoxiously long necks, your name is Steve.
D. If you do not like drawing giraffes with obnoxiously long necks, your name is not Steve.
E. NOTA
9. What is the sum of all the degrees of the interior angles in a 23-sided convex polygon?
A. 3780
B. 3960
C. 4140
D. 4500
E. NOTA
January Regional
Geometry Individual Test
10. Find the centroid of a triangle with vertices on the coordinates, (21, -9), (-7, 3) and (14, -6).
A. (9, -4)
B. (28/3, 0)
C. (28/3, -4)
D. (9, 4)
E. NOTA
11. How many of the following are true concerning rhombuses?
I) Diagonals perpendicularly intersect each other
II) Diagonals bisect two angles
III) Opposite angles sum to 180 degrees
IV) Opposite angles are congruent
V) Diagonals bisect each other
VI) Opposite sides are congruent
A. 3
B. 4
C. 5
E. NOTA
D. 6
12. Which of the following is not illustrated by the edges and vertices of
the rectangular pyramid shown on the right?
A. 3 collinear points
B. 3 non-coplanar points
C. A pair of parallel line segments
D. A pair of skew line segments
E. NOTA
13. Given that AB || CD; BC bisects
ACD; the lines AC,
CD and BC intersect at point C; the angle marked as 40°;
what is the value, in degrees, of w?
A. 70
B. 75
C. 100
D. 110
E. NOTA
w
B
A
40°
C
D
14. What is the name of the lines that meet at a point equidistant from all three sides of a triangle?
A. Angle bisector
B. Perpendicular bisector
C. Altitude
D. Median
E. NOTA
15. Which of the following does not represent the side lengths of a possible right triangle?
A. 7, 24, 25
B. 5, 12, 13
C. 20, 21, 29
D. 11, 60, 61
E. NOTA
16. Within a regular hexagon, a square is inscribed with side lengths 6. What is the length of one side of the
hexagon?
A. 3 + √3
B. 6 - √3
C. 2 + 3√3
D. 4 - √3
E. NOTA
17. Which of the following is not a valid theorem for proving similar triangles?
A. AA
B. SSA
C. SAS
D. SSS
18. A midsegment is drawn on a trapezoid with parallel bases with lengths
10 and 23. What is the length of the z, the center-most segment among the
trisected pieces of the midsegment?
A. 6.5
B. 8
C. 11
D. 13
E. NOTA
E. NOTA
23
z
10
January Regional
Geometry Individual Test
19. In a 72-sided regular convex polygon, what is the measure (in degrees) of one of the angles?
A. 160
B. 167.5
C. 170
D. 175
E. NOTA
20. Given the triangle as diagramed to the right with side lengths and
angle measures as indicated, what is the length of the side marked y?
A. 23/2 + 23√3/2
B. 46
C. 23√2 + 23√3/2
D. 23√3 – 23/2
E. NOTA
21. What is the cosine of an angle in an equilateral triangle?
A. ½
B. √2/2
C. √3/2
75°
23
60°
y
D. 1
E. NOTA
22. i, j and k represent the side lengths of a right triangle with side length i as the hypotenuse, while h
represents altitude drawn to the hypotenuse. What is i written in terms of j, k and h?
A. (j + k)/h
B. jk/h
C. √(j2 + k2)/h
D. 2(j + k)/h
E. NOTA
23. Given that ΔCAT and ΔDOG are similar triangles and line segments CA, AT, DG and DO have
respective side lengths 3, 7, 4 and 5, what is semi-perimeter of ΔDOG?
A. 31/5
B. 31/3
C. 62/5
D. 62/3
E. NOTA
24. A square is inscribed inside an equilateral triangle with side lengths 2.
What is the length of one side of the square?
A. 3√3 – 3
B. 2√3 – 4
C. 4√3 – 6
D. 6√3 – 9
E. NOTA
25. In ΔCOW, sides CO and CW have lengths of 5 and 4 respectively. Angle OCW is bisected by line
segment CD with point D lying on the line segment OW, which has length 18. What is length of segment
DO?
A. 7
B. 8
C. 9
D. 10
E. NOTA
26. In a right triangle, two of the sides have measures 3 and 3√2, what is the product of all possible values for
the unknown side length.
A. 9√2
B. 9√3
C. 27
D. 12√2
E. NOTA
27. What is the distance between the coordinates (3, 23) and (-17, 44).
A. 23
B. 25
C. √823
D. 29
E. NOTA
28. What is the maximum number of sections that can be formed when a circle is divided by 23 chords?
A. 230
B. 242
C. 253
D. 277
E. NOTA
29. Given an isosceles triangle Δ RAM with vertex R and a square WORM; line segments RA and AM which
have lengths 3 and 4 respectively, what is the perimeter of the pentagon WORAM?
A. 16
B. 17
C. 19
D. 23
E. NOTA
A
30. In the diagram to the right, G and O are right angles. If lengths TO, RT and GT
are 3, 5 and 23 respectively, what is the length of side GA?
G
A. 69/4
B. 92/3
T
C. 115/3
D. 115/4
O
E. NOTA
R