2013 Student Delegates Statewide_ _________________Geometry Individual Test
For all questions, choice (e) NOTA means none of the other given answers are correct. Diagrams are
not necessarily drawn to scale. All angle measures on this test are in degrees; disregard all other
units. All objects on this test are nondegenerate cases.
1. The cube ABCDEFGH is pictured to the right.
Which of the following points are coplanar?
a)
b)
c)
d)
e)
A, B, C, E
C, D, E, F
B, C, G, H
A, C, D, G
NOTA
2. Find the distance between the points (-7, 12) and (8, -8) in the coordinate plane.
a) 15 5
b) 35
c) 28
d) 10 3
e) NOTA
3. Find the length of an altitude of an equilateral triangle with side length 12.
a) 18 5
b) 24
c) 6 3
d) 12 2
e) NOTA
4. Exactly how many of the following are true?
I.
Through two (distinct) points there exists exactly one line.
II.
Through three (distinct) noncollinear points there exists exactly one plane.
III.
The sum of the measures of the interior angles of a triangle is 180 degrees.
IV. All squares are rectangles.
a) One
b) Two
c) Three
d) All four
e) NOTA
5. A heptagon has seven sides. Find the number of diagonals in a heptagon.
a) 21
b) 42
c) 28
d) 14
e) NOTA
6. Find the sum of the measures of the interior angles of an octagon in degrees.
a) 360
b) 900
Last saved August 5, 2012 10:50 PM by Jeremy Liu Page 1 of 5 c) 1080
d) 1440
e) NOTA
2013 Student Delegates Statewide_ _________________Geometry Individual Test
7. Trapezoid ABCD has AB || CD, AB = 10, BC = 8, CD = 16, and DA = 8.
Find the height of ABCD.
55
a)
c) 10
b) 8 3
d) 12
e) NOTA
8. ABC is a triangle. M is on segment AB and N is on segment AC such that MN is parallel to
BC. If AM = 6, MB = 10, and AN = 8, find the length of segment NC.
a)
!"
!
b) 22
c)
!"
!"
!
d)
!"
!
e) NOTA
!"
9. M and N lie on line AB and !" = !" = 3. If MN = 18, find the length of segment AB.
a) 6
b) 12
c) 24
d) 15
e) NOTA
10. A circle with center O and radius r has chords AB and CD which intersect at P.
If OP = 2, AP = 4, and PB = 6, find the area of the circle.
a) 18π
b) 25π
c) 28π
d) 30π
e) NOTA
11. The points A, B, and C are in the plane such that AB = 11, BC = 30, and CA = 32.
Which of the following is correct about A, B, and C?
a) They form an
acute triangle.
b) They form a
right triangle.
c) They form an
obtuse triangle.
12. In the diagram, the lines l and m are parallel and
intersected by a third line.
Find the measure of angle θ in degrees.
a)
b)
c)
d)
e)
78
48
156
138
NOTA Last saved August 5, 2012 10:50 PM by Jeremy Liu Page 2 of 5 d) They are
collinear.
e) NOTA
2013 Student Delegates Statewide_ _________________Geometry Individual Test
13. A triangle has sides of length 10, 10, and 16.
Find the length of this triangle’s median to a side of length 10.
(Hint: You do not need Stewart’s Theorem!!)
a) 4 10
b) 3 17
c) 2 39
d) 18 2
e) NOTA
d) 138
e) NOTA
14. Find the area of a triangle with side lengths 21, 20, and 13.
a) 120
b) 126
c) 132
15. In triangle ABC, D is on segment BC such that AD
bisects angle BAC. If AB = 24, BD = 8, AC = 15, find
the length of segment CD. (Hint: Draw CE, where E is
on AD and CE || AB, as in the diagram.)
a)
b)
c)
d)
e)
3
5
7
9
NOTA
16. ∆ABC has side lengths AB = 5, BC = 29, and AC = 30. Let G be the centroid of ∆ABC.
Given that the area of ∆ABC is 72, find the distance from G to the line AC.
a) 33/20
b) 13/8
c) 5/3
d) 8/5
e) NOTA
17. Find the volume of a rectangular prism with three faces having areas 8, 9, and 12.
a) 30
b) 12 6
c) 18 3
d) 15 2
e) NOTA
18. The altitude to the hypotenuse of a right triangle splits the hypotenuse into two segments of
lengths 6 and 24. Find the length of this altitude.
a) 6
b) 9
c) 12
d) 15
e) NOTA
19. A triangle has two sides with lengths 7 and 14.
If the length of the third side is m, which of the following is necessarily true?
a) 14 < m < 17
b) 7 < m < 21
Last saved August 5, 2012 10:50 PM by Jeremy Liu Page 3 of 5 c) 7 < m < 7 5
d) 21 < m < 28
e) NOTA
2013 Student Delegates Statewide_ _________________Geometry Individual Test
20. In triangle ABC, angle B is a right angle, AB = 3, and BC = 4.
M is on segment AB such that AM = 1 and MB = 2.
N is the midpoint of BC. AN and CM intersect at P.
Find the area of triangle APC.
a)
b)
c)
d)
e)
3/2
2
5/2
1
NOTA
21. Exactly how many of the following statements are true?
i) If a quadrilateral’s diagonals bisect, then it is a parallelogram.
ii) The converse of (i)
iii) The inverse of (i)
iv) The contrapositive of (i)
a) One
b) Two
c) Three
d) All four
e) NOTA
22. A triangle has side lengths 3, 3, 2 3. Find the length of the angle bisector to the side with
length 3. (Hint: Is there anything special about this triangle?)
a)
!
!
3
b)
!
!
c) 2
d)
!
!
+ 3
e) NOTA
23. Triangle ABC has AB = 9, BC = 26, and CA = 27.
D is the point of tangency of the incircle of ABC with BC. Find BD.
a) 3
b) 5
c) 7
d) 9
e) NOTA
24. Exactly how many of the following statements are true?
I.
If two triangles share two side lengths and the measure of the angle between those
two sides, then the triangles are congruent.
II.
If one triangle is similar to another, and the first triangle has twice the perimeter of
the second, then the first triangle also has twice the area of the second.
III.
All right triangles are similar.
IV. In quadrilaterals ABCD and WXYZ, if AB = WX, BC = XY, CD = YZ, and DA =
ZW, then ABCD is congruent to WXYZ. (In other words, there is SSSS congruence
for quadrilaterals.)
a) One
b) Two
Last saved August 5, 2012 10:50 PM by Jeremy Liu Page 4 of 5 c) Three
d) All four
e) NOTA
2013 Student Delegates Statewide_ _________________Geometry Individual Test
25. Which of the following is equal to the length of segment AB
in the diagram? (Hint: Test each answer choice.)
a)
b)
c)
d)
e)
10 3
50 3 − 50 sin 40°
50 3 − 50 tan 40°
50 3 − 50 tan 50°
NOTA
26. A triangle has side lengths 25, 25, and 30.
The distances from the incenter of the triangle to each of the sides are all equal to r. Find r.
a) 8
b)
!"
!
c)
!!
!
d)
!"
!
e) NOTA
27. Triangle ABC has angles A = 34, B = 77, and C = 69.
If H is the orthocenter of ABC, find the measure of angle BHC in degrees.
a) 120
b) 146
c) 132
d) 154
e) NOTA
28. Quadrilateral ZACH has perpendicular diagonals and ZA = 3, AC = 4, CH = 5. Find HZ.
a) 3 2
b)
!
c)
!
!
!
d)
!!
!
e) NOTA
29. Daniel draws a regular octagon and n of its diagonals.
Find the minimal value of n which guarantees that two diagonals in his drawing are parallel.
a) 2
b) 6
c) 9
d) 13
e) NOTA
30. ABCD is a square with side length 1. If E and F are on the segments BC and CD respectively
such that triangle AEF is equilateral, find the side length of triangle AEF.
a)
!
!
2
b)
6− 2
Last saved August 5, 2012 10:50 PM by Jeremy Liu Page 5 of 5 c) 2 + 3
d)
!
!
e) NOTA