Geometry Problems For Math Competitions

317 Pages·5.725 MB·English

by  Jerry Gu

Geometry Problems for
Math Competitions
Jerry Gu
Deep Path
LLC
Preface
“Mathematics is the language with which God has written the universe”
- Galileo Galilei
Especially when learned during one’s formative years, mathematics can serve
as a foundation for success in many areas. Learning mathematics allows us to
decipher the world around us through pattern recognition and logical reasoning.
In particular, the subject of geometry is a powerful conduit toward the teaching
of critical thinking and problem-solving skills.
This book is aimed toward middle- and high-school students who enjoy par-
ticipating in math contests and want to master the topic of geometry. In this
approach, I have tried to make learning feel effortless and exciting by providing
concise knowledge sections coupled with thought-provoking practice problems.
This book has two main sections: a theory section with 6 units and a prac-
tice section with 225 problems. Each problem has several detailed solutions
that contain diagrams to aid in the students’ visualization process. The book
also provides a hint box below each question that gives guidance toward the
recommended method of solving the problem, in case of difficulty.
I wrote this book based on my personal experience from participating in math
competitions, and the topics I’ve noticed students commonly struggle with. I
hope that by the end of the book, readers will feel more confident in their critical
thinking abilities and further their appreciation for geometry.
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Acknowledgements
I want to start by thanking Mrs. Porterfield, who has served as our high
school’s Mathcounts coach for many years. She made it her mission to afford
all of her students an opportunity to succeed in math. I am grateful for the two
years I spent in her class, where she introduced me to many middle and high
school math contests, which helped fuel my passion for mathematics. Thank you
for the time and effort you spent coaching Mathcounts early in the mornings
while the sky was still dark; I appreciate the countless hours you put into
organizing contests.
I cannot thank my amazing parents, Yong Gu and Lily Qiao, enough for
their unconditional love and support. My father is a first generation immigrant
who grew up in extreme poverty in rural China. During his childhood, he had
an insatiable curiosity that led him to learn something new whenever he was
given the opportunity. His family could not afford to buy books for him, so
he spent hours everyday self studying in bookstores and public libraries. My
father’s passion for learning, despite his circumstances, paid off when he placed
highly in the Chinese National Math Olympiad and was admitted to one of the
best universities of China with a scholarship. He became the first in his family
to attend college. My mother, a rocket scientist and professor, is also the first
in her family to obtain a college degree. Her love of science and her passion of
teaching and mentoring students inspire me every single day. The story of my
parents’ success has motivated me to work hard to realize my dreams. I would
also like to thank my creative, artistic, 7-year old sister Emily for giving me
ideas of the front cover design.
I would like to thank the following contests for providing me with their
problems:
The MATHCOUNTS Series is a premier middle school math competition
that aims to help students develop a passion for mathematics, as well as build
confidence and critical thinking skills.
The American Math Competitions (AMC 10, AMC 12) is a series of exam-
inations for middle and high school students that build problem-solving skills
and mathematics knowledge.
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Contents
1 An Introduction 5
2 Fundamental Knowledge 6
2.1 Lines and Angles . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Angles from intersecting lines . . . . . . . . . . . . . . . 6
2.1.2 Angles from parallel lines . . . . . . . . . . . . . . . . . 6
2.1.3 Angles from perpendicular lines . . . . . . . . . . . . . 7
2.2 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Types of triangles . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Altitudes, medians, midlines, angle and perpendicular bi-
sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Isosceles and equilateral triangles . . . . . . . . . . . . . 11
2.2.4 Congruent and similar triangles . . . . . . . . . . . . . . 12
2.2.5 The Pythagorean theorem . . . . . . . . . . . . . . . . 13
2.2.6 Right triangles . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.7 Inequalities in a triangle . . . . . . . . . . . . . . . . . 15
2.2.8 Area of a triangle . . . . . . . . . . . . . . . . . . . . . 16
2.2.9 The Area method, Ceva’s Theorem, and Menelaus’ Theorem 16
2.2.10 The Law of Sines and Cosines . . . . . . . . . . . . . . 18
2.3 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Circumference and area . . . . . . . . . . . . . . . . . . 19
2.3.2 The diameter of a circle . . . . . . . . . . . . . . . . . . 20
2.3.3 Tangent lines to circles . . . . . . . . . . . . . . . . . . 20
2.3.4 Inscribed angles in a circle . . . . . . . . . . . . . . . . 22
2.3.5 Inscribed and circumscribed circles . . . . . . . . . . . . 23
2.3.6 The Power of a Point Theorem . . . . . . . . . . . . . . 23
2.3.7 Ptolemy’s Theorem . . . . . . . . . . . . . . . . . . . . 24
2.3.8 Concyclic (or cocyclic) points . . . . . . . . . . . . . . . 24
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2.4 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Parallelograms . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2 Trapezoids . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Pentagons . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.4 Hexagons and other polygons . . . . . . . . . . . . . . . 28
2.5 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.1 Cubes and rectangular prisms . . . . . . . . . . . . . . 28
2.5.2 Triangular pyramids and square pyramids . . . . . . . . 29
2.5.3 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.4 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.5 Frustums . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Analytical Geometry . . . . . . . . . . . . . . . . . . . . . . . 32
2.6.1 Coordinate system basics . . . . . . . . . . . . . . . . . 32
2.6.2 Lines in coordinate system . . . . . . . . . . . . . . . . 33
2.6.3 Circles in coordinate system . . . . . . . . . . . . . . . 34
2.6.4 The distance formula . . . . . . . . . . . . . . . . . . . 35
2.6.5 Geometry of Intersections . . . . . . . . . . . . . . . . . 36
2.6.6 Symmetry in coordinate system . . . . . . . . . . . . . 37
2.6.7 Graphing absolute values . . . . . . . . . . . . . . . . . 39
3 Two-hundred-twenty-five Problems 41
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Chapter 1
An Introduction
Geometry is a branch of mathematics focused on the study of the sizes, positions,
angles, and dimensions of an object. The word geometry originates from the
Greek word geometria - to measure the Earth. In ancient Greece, the scholar
Eratosthenes used geometry to be the first person to measure the diameter of
the Earth. Since being founded by Euclid in Alexandria, Egypt in 300 BC,
geometry has been applied throughout history to build grandiose structures
such as the Colosseum in Rome, the Taj Mahal in India, and even modern
skyscrapers such as the Burj Khalifa in Dubai.
If you look hard enough, you will see that geometry occurs all around you.
From the beautiful hexagonal prisms found in snowflakes to the fractals found
in Romanesco Broccoli, which form beautiful spirals that follow the Fibonacci
sequence, nature is full of geometry. In this book, you will uncover the secrets
and gain a deep appreciation of geometry.
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Chapter 2
Fundamental Knowledge
2.1 Lines and Angles
2.1.1 Angles from intersecting lines
Angles are generated through the crossing of two lines.
(1) Angles 1 and 2 (left diagram) are linear angles and they are supplementary, meaning that the
sum of the angles is 180◦.
(2) Angles 1 and 2 (right diagram) are called vertical angles. Vertical angles are congruent.
2.1.2 Angles from parallel lines
Three types of angle pairs are created when two parallel lines are crossed by a transversal line:
corresponding angles, alternate interior angles, and alternate exterior angles. The angles in these
angle pairs are congruent and its converse is also true.
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(1) Corresponding angles: 1&5, 2&6, 3&7, 4&8
(2) Alternate interior angles: 3&6, 4&5
(3) Alternate exterior angles: 1&8, 2&7
2.1.3 Angles from perpendicular lines
Perpendicular lines create 90º angles.
(1) ∠1+∠2 = 90◦
2.2 Triangles
2.2.1 Types of triangles
(a) Acute, right and obtuse triangles
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(b) triangle and equilateral triangle
2.2.2 Altitudes, medians, midlines, angle and perpendicular bisec-
tors
(a) Altitudes of a triangle
(1) In ∆ABC, CD is the altitude to side AB.
(2) The area of ∆ABC is equal to ch/2.
(3) Given a, b and c as the side lengths of a triangle, Heron’s formula can be used to calculate
the area. A = (cid:112)s(s−a)(s−b)(s−c), where s = (a+b+c)/2. So the altitude h = 2A/c.
(4) Given the side lengths of a triangle as a, b and c, using the Pythagorean Theorem yields
(cid:112)
a2−b2 = y2−x2 = c(y−x) and x+y = c. Thus y = (a2+c2−b2)/2c and h = a2−y2.
(5) Given the side lengths of a triangle as a, b and c, then we know h = bsinA. If the measure of
angle A is not given, however, we must use the Law of cosines to find cos A, which results
in cosA = (b2+c2−a2)/(2bc). Then the identity (sinA)2+(cosA)2 = 1 can be used to solve
for sin A.
(b) Medians of a triangle
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(1) In ∆ABC, if D is the midpoint of segment AB, then CD would the median the side AB.
(2) The three medians of the triangle , CD, AE and BF, meet at a the centroid G, which is the
center of gravity.
(3) The distance from the centroid to each vertex is 2/3 the length of each median. CG = 2GD,
AG = 2GE and BG = 2GF.
(4) The area of each small triangle is 1/6 of the area of ∆ABC.
(5) The Median Length Theorem: 2BD2 = AB2+BC2−AD2−BD2
(c) Midlines of a triangle
(1) Lines DE, EF and FD are midlines of ∆ABC when D, E and F are midpoints of side AB,
BC and CA, respectively.
(2) FE//AB, DE//AC and DF//BC.
(3) All of the triangles in the diagram are similar to each other.
(4) The area of each small triangle is 1/6 of the area of ∆ABC.
(d) Angle bisectors of a triangle
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