FGCU Invitational
Geometry Individual
2015
Directions: For all questions, NOTA represents none of the above answers is correct.
1)
If the line containing points (3,1) and (−5, 𝑚) is parallel to the line containing points
(6, 𝑚 + 1) and (11,0), what is the value of 𝑚?
a) − 13
3
d) − 6
7
b) − 1
4
e) NOTA
c)
2)
The measure of each interior angle of a regular polygon is nine times that of an exterior
angle of the polygon. Determine the number of sides of the polygon.
a) 28
d) 24
b) 20
e) NOTA
c)
3)
16
The interior angles of a pentagon are 3𝑥, 3𝑥 + 24, 3𝑥 + 48, 3𝑥 + 72, and 3𝑥 + 96.
What is the measure of the smallest angle?
a) 24°
d) 36°
b) 18°
e) NOTA
c)
4)
2
−5
20°
Segment 𝑂𝐵 bisects angle ∠AOC. If 𝑚∠𝐴𝑂𝐵 = 2𝑥 + 5 and 𝑚∠𝐵𝑂𝐶 = 4𝑥 − 15, find
𝑚∠𝐴𝑂𝐶.
a) 42°
d) 55°
b) 20°
e) NOTA
c)
25°
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FGCU Invitational
5)
Geometry Individual
2015
In a triangle, one angle is three times as large as the other and the third is 100° greater than
the sum of the other two. What are the measures of the angles of the triangle?
a) 5° , 15° , 160°
d) 25° , 75° , 80°
b) 10° , 30° , 140°
e) NOTA
c)
6)
In the figure to the right, an equilateral triangle and an isosceles triangle
share a common side. What is the measure of ∠𝐴𝐵𝐶?
a) 52°
d) 112°
b) 128°
e) NOTA
c)
7)
20° , 60° , 100°
60°
In the figure, PQ = QR = RS. Find TU.
a) 7.2
d) 6.4
b) 5.4
e) NOTA
c)
4.2
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FGCU Invitational
8)
a) 2.5
d) 5
b) 2
e) NOTA
10
Find the values of 𝑥 and 𝑦.
a) 𝑥 = 15, 𝑦 = 26
d) 𝑥 = 25, 𝑦 = 40
b) 𝑥 = 35, 𝑦 = 15
e) NOTA
c)
10)
2015
Use the given figure to find the value of 𝑥.
c)
9)
Geometry Individual
𝑥 = 45, 𝑦 = 20
Given the following conditional statement:
𝐼𝑓 𝑎 𝑞𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 ℎ𝑎𝑠 𝑓𝑜𝑢𝑟 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠, 𝑡ℎ𝑒𝑛 𝑖𝑡 𝑖𝑠 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒
Determine the truth value of the inverse, contrapositive, and converse, respectively.
a) False, True, False
d) True, False, True
b) False, False, False
e) NOTA
c)
True, True, False
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FGCU Invitational
11)
a) 36
d) 40
b) 28
e) NOTA
44
Refer to the figure below with 𝑋 the midpoint of side
𝑀𝑁, 𝑌 the midpoint of side 𝑁𝐷, 𝑋𝑌 = 5𝑥 − 8, and
𝑀𝐷 = 8𝑥 − 8. Find 𝑥.
a) 8
d) 12
b) 13
e) NOTA
c)
13)
2015
The lengths of the sides of a triangle are in the extended ratio 7 ∶ 9 ∶ 10. The perimeter
of the triangle is 104 cm. What is the length of the longest side?
c)
12)
Geometry Individual
4
In the figure shown, find 𝐶𝐷?
a) 10
d) 26
b) 15
e) NOTA
c)
17
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FGCU Invitational
14)
a) 2
d) 4.5
b) 2.5
e) NOTA
3
Let 𝐷 be any point on the base of an isoceles triangle
𝐴𝐵𝐶. Extend 𝐴𝐶 to 𝐸 so that 𝐶𝐷 = 𝐶𝐸. Extend 𝐸𝐷 to
meet 𝐴𝐵 at 𝐹. If angle 𝐶𝐸𝐷 is 10 degrees, find angle
𝐴𝐹𝐷.
a) 20°
d) 50°
b) 30°
e) NOTA
c)
16)
2015
In the figure, 𝐴𝐶 is 21 units, 𝐴𝐷 is 𝑥 units, 𝐷𝐵 is 5
units, 𝐷𝐸 is 7 units, 𝐵𝐸 is 𝑥 units and ∠𝐵𝐷𝐸 = ∠𝐵𝐶𝐴.
What is the value of 𝑥.
c)
15)
Geometry Individual
40°
The area of a square inscribed in a semicircle is to the area of the square inscribed in
the entire circle as:
a) 1 ∶ 2
d) 2 ∶ 3
b) 3 ∶ 4
e) NOTA
c)
2∶5
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FGCU Invitational
17)
b)
c)
𝑒−𝑏
𝑎
𝑎
𝑒−𝑏
𝑎
𝑏−𝑒
e) NOTA
𝑏−𝑒
𝑎
ℎ𝑑
2ℎ+𝑑
b) ℎ + 𝑑 − √2𝑑
c)
d) √ℎ2 + 𝑑 2 − ℎ
e) NOTA
𝑑−ℎ
In the square 𝐴𝐵𝐶𝐷, the point 𝐸 is between 𝐴 and 𝐵 and the point 𝐹 is between 𝐵 and
𝐶. Find ∠𝐴𝐸𝐹 + ∠𝐸𝐹𝐶.
a) 90°
d) 360°
b) 180°
e) NOTA
c)
20)
d)
In the right triangle shown, the sum of 𝐵𝑀 and 𝑀𝐴 is equal to
the sum of 𝐵𝐶 and 𝐶𝐴. If 𝑀𝐵 = 𝑥, 𝐶𝐵 = ℎ, and 𝐶𝐴 = 𝑑, find 𝑥.
a)
19)
2015
𝐴(3𝑎, 3𝑏), 𝐵(3𝑐, 3𝑑), and 𝐶(0,3𝑒) are the vertices of a triangle. Find the slope of the
altitude from 𝐵 to 𝐴𝐶.
a)
18)
Geometry Individual
270°
The parabolas 𝑦 = −4𝑥 2 + 3𝑥 + 5 and 𝑦 = 2𝑥 2 + 4𝑥 − 7 intersect in the points 𝑃 and
𝑄. What is the slope of the straight line through 𝑃 and 𝑄.
a) − 3
11
b)
11
c)
3
3
11
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d) − 11
3
e) NOTA
FGCU Invitational
21)
2015
𝑘
The points (4,3), (2, −3), and (5, 2) are on the same straight line. Find the value(s) of
𝑘.
a) 6
d) 6,12
b) 12
e) NOTA
c)
22)
Geometry Individual
−8, −8
The points 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸 are located on a straight line, in order, in accordance with
the following conditions. What is the distance from 𝐵 to 𝐶?
The distance from 𝐴 to 𝐸 is 40cm.
The distance from 𝐴 to 𝐷 is 35cm.
The distance from 𝐵 to 𝐸 is 20cm.
𝐶 is halfway between 𝐵 and 𝐷.
a) 6.5 cm
d) 7.5 cm
b) 5 cm
e) NOTA
c)
23)
2.5 cm
What is the value of the following product?
cot 5° ∙ cot 15° ∙ cot 25° ∙ cot 35° ∙ cot 45° ∙ cot 55° ∙ cot 65° ∙ cot 75° ∙ cot 85° ∙
a) 0
d) 1.5
b) 0.5
e) NOTA
c)
1
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FGCU Invitational
24)
25)
a)
30
b)
10
c)
50
11
3
d)
50
9
e) NOTA
13
If the graphs of the equations 5𝑥 − 3𝑦 = 13 and 4𝑥 + 𝑎𝑦 = 11 intersect at right
angles, find the value of 𝑎.
b)
c)
27)
2015
Parallelogram 𝑃𝑄𝑅𝑆 has 𝑃𝑄 = 𝑅𝑆 = 8 𝑐𝑚, and diagonal 𝑄𝑆 = 10 𝑐𝑚. Point 𝐹 is on
segment 𝑅𝑆, exactly 5 𝑐𝑚 from 𝑆. Let 𝑇 be the intersection of segments 𝑃𝐹 and 𝑄𝑆.
Find the length of 𝑇𝑆.
a) − 20
3
26)
Geometry Individual
3
5
d) − 5
3
e) NOTA
5
12
Compute the perimeter of the equilateral triangle with an area of 1.
1
a)
4
b)
4
c)
4
√3
2
√3
d)
6
4
√3
e) NOTA
4
√3
Square 𝐴𝐵𝐶𝐷 has a side length of 6 and two equilateral triangles 𝐴𝐵𝐸 and 𝐴𝐵𝐹 are
drawn such that 𝐸 is on the interior of 𝐴𝐵𝐶𝐷 and 𝐹 is on the exterior of 𝐴𝐵𝐶𝐷.
Determine the area of triangle 𝐶𝐹𝐸.
a) √3
d) 18√3
b) 3√3
e) NOTA
c)
9√3
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FGCU Invitational
28)
29)
2015
Consider square 𝐴𝐵𝐶𝐷 with side length 1. Isosceles triangle 𝐴𝐵𝐸 where 𝐴𝐸 = 𝐵𝐸 is
drawn on the exterior of the square such that the lengths 𝐶𝐷, 𝐴𝐶, and 𝐶𝐸 are in
increasing geometric progression. Determine the area of 𝐴𝐵𝐸.
1
a)
√15
4
−2
d)
b)
√15
4
−1
e) NOTA
c)
√15
4
√15
2
Trapezoid 𝐴𝐵𝐶𝐷 has an inscribed circle as shown. 𝐸𝐻 is
the median of the trapezoid, and points 𝐹 and 𝐺 are on
the inscribed circle, and are inside the trapezoid. If 𝐸𝐻 =
10 and the enclosed area of the trapezoid is 70, find the
value of 𝐸𝐹 = 𝐺𝐻.
a) 10 − 4√3
d) 2
b) 5 − 2√3
e) NOTA
c)
30)
Geometry Individual
3
Find the value of 𝑥.
a) 121
d) 76
b) 100
e) NOTA
c)
112
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