Download Test

Geometry Individual – January 16, 2016 – FAMAT State-Wide Competition hosted by Vero Beach HS
1. Looking at the diagram to the right, you see the median plane of the bow shown.
Traditionally, in a triangle, the median of a triangle is the line segment that connects:
_______.
A. Two vertices of a triangle
D. A midpoint to the
B. A vertex and the midpoint of the opposite side
opposite side
C. A vertex to the opposite side at a right angle
E. None of these Answers
2. From Robin Hood to Legolas, there have been many iconic archers in literature and
film: at the Olympic Games you can see all this skill and heroism in real life. Archery
made its Olympic debut in 1900. The simple goal is to hit a target 70 meters away.
Hitting a target 70 meters away scores a point. The simple goal is to score a point.
What is the name of this logic used in bold?
A. Law of Syllogism
D. Law of
B. Reflexive Property
Detachment
C. Symmetric Property
E. None of these Answers
3. The paths of two arrows are intersecting lines. At the point of intersection, two vertical angles are formed. Their
measures are (4𝑥 − 8)° and (7𝑥 − 50)°. Find the sum of the vertical angles.
A. 28°
D. 96°
E. None of these Answers
B. 48°
C. 56°
4. Given the right triangle formed in the diagram below, with the right angle at the bow,
calculate the distance from the bow to the target if the center of the bow is 60 cm
above the ground and the distance from his shoes to the target is 61 cm.
The bow is directly above his shoes.
A. 6 cm
D. 21 cm
B. 11 cm
E. None of these
C. 16 cm
Answers
5. Everett wants to enlarge one of his triangular targets that he uses for practice.
Use the diagram to find the value of x of the target and by what property can
you find it?
A. 323⁄8 by AA Similarity
B. 323⁄8 by AAS Similarity
C. 51⁄8 by AA Similarity
D. 51⁄8 by AAS
Similarity
E. None of these
Answers
1
Geometry Individual – January 16, 2016 – FAMAT State-Wide Competition hosted by Vero Beach HS
6. Write a converse to the following statement: If you are New Zealand archer Neroli Fairhallm, then you became
the first paraplegic athlete to compete in the Olympic Games in 1984.
A. If you do not became the first paraplegic athlete to compete in the Olympic Games in 1984, then you are not New
Zealand archer Neroli Fairhallm.
B. If you are not New Zealand archer Neroli Fairhallm, then you became the first paraplegic athlete to compete in the
Olympic Games in 1984.
C. If you are not New Zealand archer Neroli Fairhallm, then you will not became the first paraplegic athlete to
compete in the Olympic Games in 1984.
D. If you became the first paraplegic athlete to compete in the Olympic Games in 1984, then you are New Zealand
archer Neroli Fairhallm.
E. None of these Answers
7. Everett is using a convex polygon target for archery practice. He is
interested in solving for the second largest (only one larger) exterior angle
the polygon. Can you help him find the angle measure in degrees? Use the
diagram where all angle measures are measured in degrees.
A. 65°
B. 85°
C. 110°
D. 125°
E. None of these Answers
of
8. If the largest possible cross-section of the head of the arrow is an isosceles triangle with legs of length 2 mm. Use
the diagram to find the possible value(s) for the length of the base b of the arrowhead?
A. 2 < 𝑏 < 4
B. 0 < 𝑏 ≤ 2
C. 0 < 𝑏 ≤ 4
D. 0 < 𝑏 < 4
E. None of these Answers
9. Everett is aiming his bow and arrow at a target
labeled with a Cartesian Plane. His four arrows land on the following points: (−7,4) , (−2,16) , (−3,5) , (2,17).
What term is the best description of the shape formed by those four points in the order stated?
A. Trapezoid
B. Quadrilateral
C. Square
D. Parallelogram
2x+5
E. None of these Answers
5y+15
10. Using the parallel lines with angle measures given,
find the sum of x and y. Not drawn to scale.
A. 25
B. 50
C. 75
3y+5
D. 100
E. None of these Answers
11. Historically archery has been used for hunting and combat. But in the Olympics, it is all about
precision and accuracy. The steps to shooting a bow and arrow are as follows: determine your
dominant eye, select an arm guard and chest protector, assume the shooting stance
2
Geometry Individual – January 16, 2016 – FAMAT State-Wide Competition hosted by Vero Beach HS
perpendicular to the target, pull back the string, aim and shoot. In this process, can you name the objects that are
MOST similar to a point, line or line segment, and plane? (in that order)
A. Eye, string, chest protector
D. String, Eye, Arm Guard
B. Eye, chest protector, string
E. None of these Answers
C. String, arm guard, Eye
12. At the opening ceremony for the Barcelona 1992 Olympic Games, Spanish Paralympic archer Antonio Rebollo lit
the cauldron by shooting a flaming arrow into it. Using this information, I write the following algebraic statement:
If 1992 = 𝑥, then 𝑥 = 1992. What property am I using?
A. Transitive Property
B. Symmetric Property
D. Equality Property
C. Reflexive Property
E. None of these Answers
13. In an archery competition, two rectangular shooting ranges (the areas where the action happens) are known to be
similar. For the smaller of the rectangles, the width is known to be three more than twice the length. The
perimeter of the largest rectangle is known to be 20 units. If the scale factor of the rectangles is known to be 4:7,
find the width of the smaller rectangle.
A. 181⁄7
D. 181⁄17
E. None of these Answers
B. 101⁄21
C. 71⁄21
14. A diagram of an index feather of an arrow has been enlarged and displayed to
the right. Based on the diagram as marked, which of the following is false?
A. 𝐴𝐶 ≅ 𝐵𝐷 by AAS Congruence and CPCTC
B. 𝐶𝐵 ≅ 𝐵𝐶 by Reflexive Property
C. ⊿𝐴𝐶𝐵 ≅ ⊿𝐷𝐵𝐶 by AAS Congruence
D. 𝑚∠𝐴 ≅ 𝑚∠𝐷 by Given Information
E. None of these Answers
15. When drawing a bow (pulling the string back to shoot) the best form is when the line through the center of your
body and hand are both perpendicular to the arrow.
Which of the following could be possible equations for
the lines of the archer’s body, hand, and arrow
respectively?
A. 2𝑥 − 3𝑦 = 12; 2𝑥 − 3𝑦 = 8; 2𝑥 − 3𝑦 = 3
B. 3𝑥 + 2𝑦 = 12; 2𝑥 − 3𝑦 = 8; 3𝑥 + 2𝑦 = 3
C. 2𝑥 − 3𝑦 = 12; 2𝑥 − 3𝑦 = 8; 3𝑥 + 2𝑦 = 3
D. 3𝑥 − 2𝑦 = 12; 2𝑥 − 3𝑦 = 8; 3𝑥 + 2𝑦 = 3
E. None of these Answers
16. Archery was introduced for the first time at the Games of the II Olympiad in Paris in 1900. It was then included
on the program of the 1904, 1908 and 1920 Games before disappearing for over 50 years. The Games of the XX
Olympiad in Munich in 1972, saw the re-introduction of archery on the Olympic program. Women were able to
compete in archery events at the 1904 and 1908 Games, then again, like the men, in 1972. How many numbers in
3
Geometry Individual – January 16, 2016 – FAMAT State-Wide Competition hosted by Vero Beach HS
the paragraph above are side lengths in a Pythagorean Triple? (Include Roman Numerals and multiples of known
triples)
A. 5
B. 6
D. 8
C. 7
E. None of these Answers
17. The most decorated archer in Olympic history is Hubert Van Innis of Belgium who competed in
1900 and 1920, winning six gold and three silver medals. What geometric shape is a medal
most similar to? Use the graphic at the right for assistance.
A. Sphere
B. Cylinder
D. Triangular Prism
C. Pyramid
E. None of these Answers
18. In Olympic archery, the target is 1.22 meters in diameter, but, to the archer, standing those 70 meters
away, it appears about the size of a thumbtack (0.5 cm) held at arm's length. Assuming the thumbtack and the
target are parallel to each other, and the bottom of each is at the same height, what is an “arm’s length” to the
nearest centimeter?
A. 14
D. 1400
B. 29
E. None of these Answers
C. 290
19.
A.
B.
C.
D.
E.
Which of the following is true about the triangle to the right?
𝑚∠𝐵 = 48°
𝑚∠𝐴𝐷𝐶 = 48°
The angle marked with the question mark is named ∠𝐴𝐵𝐶
Δ𝐶𝐷𝐴 ≅ Δ𝐷𝐵𝐶
None of these Answers
20. The venue for Archery in the 2016 Rio Olympics is Sambódromo. Everett is located at a point 12 miles from
Sambódromo at an angle of 30 degrees east of north. If the roads in Brazil only run north/south or east/west, how
far will Everett have to travel to get to Sambódromo?
A. 6√3 + 6
B. 12√2
C. 12√3 + 6
D. 6√2 + 6
E. None of these Answers
4
Geometry Individual – January 16, 2016 – FAMAT State-Wide Competition hosted by Vero Beach HS
Solutions:
1B
2A
3D
4B
5A
Definition of a median.
Law of Syllogism
Vertical angles are congruent; 4x-8=7x-50 ; x=14 ; Each angle is 4(14)-8=48 degrees, so their sum is 96
degrees.
Using Pythagorean Theorem, we can find the missing side length to be 11 cm.
Two of the angles are marked congruent in the triangles. The third angle at Y we know is congruent
because it is the same angle in both triangles. But regardless, knowing two angles are equal is enough to
know all three are equal because the sum is always 180 degrees. Now that we know the triangles are
16
17
similar, we can set up a proportion to solve for x:
= ; 𝑥 = 323⁄8 by AA Similarity
16+22
6D
7B
8D
9 D or E
10 B
11 A
12 B
13 B
14 A or E
15 C
16 C
17 B
18 B
19 C
20 A
𝑥
To find the converse, you replace the p with the q in the If p, then q conditional statement.
To solve for x, I set the sum of all expressions equal to 540 degrees which is the interior of a pentagon.
540 = 25𝑥 + 40; 𝑥 = 20 By substituting x back into each expression, I find the interior angles to be:
168, 110, 102, 65, 95. That makes the exterior angles equal to 32, 70, 78, 115, and 85 degrees. The
second largest is 85 degrees.
According to the Triangle Inequality, the sum of the length of any two sides of a triangle must be greater
than the third side. 0 < 𝑏 < 4
First, I solved for the slope of each side: AB=12/5, CD=12/5, AC=1/4, BD=1/4 and found that it was a
parallelogram with opposite sides parallel. To check to see if it could be a rectangle, I verified that the
slopes of adjacent sides were NOT opposite reciprocals, therefore were NOT perpendicular. This makes
the quadrilateral a parallelogram at best.
I used the same-side interior angles of the parallel lines to solve for y first: 8y+20=180; y=20. Then I
substituted that value in to 3y+5 to find that angle to be 65 degrees. The corresponding angle of 2x+5
must be congruent. 2x+5=65; x=30. 30+20=50
Eye is similar to a point because it is one singular location. A string is like a line or line segment in that is
follows a linear pattern and has multiple points. A chest protector is like a plane in that it covers a large
surface area.
Symmetric Property
Oh, this is a tricky one! The scale factor is 4:7, so you know that the sides and the perimeters both follow
those ratios. When you solve the proportion to find the smaller rectangle’s perimeter, you get 80/7:
𝑃
4
= 7. From there you must solve for x: 6x+6=80/7, so x=19/21. The width is 2x+3; so width = 101/21.
20
The order is reversed for what it should be, but all of the information is otherwise correct.
The body and hand need to be parallel, which means they have the same slope. In standard form, you can
calculate the slope by –A/B. So any equations where the A and B match in the Ax+By=C will be parallel.
The arrow needs to be perpendicular to both, so its slope should be the opposite reciprocal of the others.
(B/A)
The numbers in the paragraph that are multiples of Pythagorean Triples are: 20, 50, 1900, 1904, 1908,
1920 and 1972. Most are multiples of 4 or 5 which are legs in a known triple.
Cylinder
122
7000
By setting up the similar triangles, we can use the proportion to solve for arm’s length:
=
;
0.5
𝑥
x=28.69, rounded to the nearest centimeter is 29.
Using the base angles of an isosceles triangle are congruent, we can find the measures of the angles to be:
CAD=66, ADC=66. Because they are a linear pair, we can find CDB to be 114 degrees. Using isosceles
triangles again, we can find the question mark to be 33 degrees.
By using your 30-60-90 special right triangle, the 12 units is opposite the 90 degree angle, so the distance
he must travel is the sum of the sides opposite the 30 and 60 degree angles. 6√3 + 6
5