Download Geometry Notes/Problems-Henry, 03-09-15 Math

The Fundamentals of Geometry
Henry Dai
“Do not worry about your difficulties in Mathematics. I can assure you mine are still
greater.”
-Albert Einstein
Main Topics we will go over:
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The Cartesian Coordinate System
Angles in polygons
Angles in circles
Lines in polygons (esp. triangles)
Area of polygons
Area of irregular shapes
Lines in circles (Power of a point)
Three dimensional geometry
Trigonometry (30-60-90 and 45-45-90)
The Building Blocks of Geometry
All of Geometry consists of the three building blocks: Points, Lines, and Planes (Not the flying
ones). But for all points, lines, and planes, there are variations. For instance, there are segments
and rays. There are 2D planes and there are 3D planes. Also, the building blocks can create
towers. Two intersecting lines form an angle. It’s important that we know the definition to the
following before we proceed:
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Segment
Endpoints
Midpoint
Line
Collinear
Angle
Sides
Vertex
Although it’s not a building block, angles are one of the most important things in geometry
(Duh!). I will show you some properties of angles.
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Right angles
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Acute angles
Obtuse angles
Complementary angles
Supplementary angles
Now let’s look at the role of angles in lines
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Alternate-interior angles
Corresponding angles
Exterior angles
Let’s look at some of the properties of angles in regular polygons
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Regular polygon
Interior angles
Exterior angles
180(𝑛−2)
o Interior angle:
360
o Exterior angle:
𝑛
𝑛
Angles are a huge part of geometry. But now let’s examine how shapes (polygons) work
exclusively. Specifically, the area of polygons.
o Area of a triangle : 𝐵𝑎𝑠𝑒 ∗ ℎ𝑒𝑖𝑔ℎ𝑡 ∗ 12 or Heron’s formula:
o Area of an equilateral triangle:
or 𝑟𝑠
𝑥 2 √3
4
o
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Area of a rectangle: 𝐿𝑒𝑛𝑔𝑡ℎ ∗ 𝑤𝑖𝑑𝑡ℎ
Area of a parallelogram: 𝐵𝑎𝑠𝑒 ∗ ℎ𝑒𝑖𝑔ℎ𝑡
o
Area of a trapezoid: (𝐵𝑎𝑠𝑒1 + 𝐵𝑎𝑠𝑒2)(ℎ𝑒𝑖𝑔ℎ𝑡)
o
Area of a rhombus:
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Area of a regular hexagon:
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# of diagonals in an n-sided polygon,
1
2
𝑑1𝑑2
2
3𝑥 2 √3
2
𝑛(𝑛−3)
2
Triangles
Every single polygon consists of smaller triangles. Triangles are an integral part of geometry and
simplifies problem solving all around. Let’s look at some types of triangles and their properties
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Acute triangle
Right triangle
Obtuse triangle
Equilateral triangle
Isosceles triangle
Legs
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Hypotenuse
Vertex angles
Base angles
Scalene
There are many variations of triangles, but there are certain attributes that every triangle has.
Let’s analyze some parts to a triangle. We can derive many formulas and simplify problems by
knowing these properties
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Medians
Centroid
Angle bisectors
Incenter
Perpendicular bisector
Circumcenter
Altitudes
Orthocenter
There are also important formulas associated with triangles
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The Triangle Inequality
Pythagorean theorem
Degenerate
Pythagorean Triples
We can use angles and sides to culminate another important concept: Congruent and Similar
Triangles.
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SSS Congruency
SAS Congruency
ASA Congruency
AAS congruency
AA Similarity
SAS Similarity
SSS Similarity
All right. Let’s take a break from triangles and polygons. If I told you that perhaps the most simplelooking figure is actually the most complicated figure math-wise, would you believe it? Let me show you
why circles appear to be so simple, but have many properties and tricks that you might want to learn.
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Inscribed angles
Central angles
Radius
Center
Diameter
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Chord
Tangent
Secant
Concentric
Arc
Minor arc
Major arc
Sector
Circumference
Application of the Pythagorean theorem
Application of Similar Triangles (Power of a Point)
Relationship with a hexagon
Equation for a circle
More concepts and formulas to remember:
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The distance formula:
General form for a circle: 𝑥 2 + 𝑦 2 + 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
Simplified form for a circle:(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
Power of a Point formulas (optional)
Formulas for 3D shapes (Volume, Surface Area, Diagonals)
Here are some problems for practice. And remember, the AOPS book contains all of this information and
a lot of good problems as well!
1. Let
(2012 AMC 10A #4)
and
. What is the smallest possible degree measure for angle CBD?
2. Find the number of diagonals that can be drawn in a polygon of 70 sides.
3. What is the measure of an interior angle of a 20-gon?
4. An equilateral triangle and regular hexagon have equal perimeters. If the area of the triangle is 1. Find the area of
the hexagon.
5. The set, S, contains the first five counting numbers. How many subsets of S can be the sides of a triangle?
6. The complement of angle, A, is 2 less than 3 times the measure of angle A. What is the measure of A?
7. The line
forms a triangle with the coordinate axes. What is the sum of the lengths of
the altitudes of this triangle? (AMC 2015)
8. In equilateral triangle, ABC, the three medians intersect at point, P. What is the length of segment AP?
9. In a circle, a chord AB has length 6. A perpendicular bisecting segment, that is shorter than the radius, intersects
AB at point C, and has length 2. What is the radius of the circle?
10. Find the area of a rhombus with a side of length 13 and one diagonal of length 24.
11. A square is inscribed inside circle H. Circle H is circumscribed about another square. What is the ratio of the
larger square to the smaller square?
12. Square ABCD is inscribed inside a circle. A smaller square, with a side on CD and a parallel side that is a chord of
the circle, has side length 2. What is the radius of the circle?
13. ABCDEF is a regular hexagon. The distance from the midpoint of CF to AB is 3. What is the area of the hexagon?
14. In trapezoid DARN, the two non-parallel sides have lengths of 13 and 20. The smaller base has length 15, and
the height has length 12. What is the area of trapezoid DARN?
15. Triangle ABC is right-angled at B. The median from B intersects AC at point D. What is the length of AD?
16. Find the centroid of a triangle with vertices (5, 11), (18, 29), (13, 26).
17. What is the area of a sector with a central angle of 30 degrees in a circle with radius 5?
18. The isosceles right triangle
has right angle at and area
. The rays trisecting
intersect
at and . What is the area of
? (AMC 2015)
19. For some positive integers , there is a quadrilateral
perimeter , right angles at and ,
, and
with positive integer side lengths,
. How many different values of
are possible? (AMC 2015)
20. In rectangle
,
and points
and
lie on
so that
. What is the ratio of the area of
to the area of rectangle
and
trisect
? (AMC 2014)
21. Two points on the circumference of a circle of radius r are selected independently and at random.
From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two
chords intersect? (AMC 2011)
22. Equiangular hexagon
has side lengths
. The area of
is
of all possible values of ? (AMC 2010)
23. Convex quadrilateral
intersect at ,
(AMC 2009)
24. Let
parallel to
has
, and
and
of the area of the hexagon. What is the sum
and
and
be a square, and let
and
be points on
and the line through
parallel to
. Diagonals
and
have equal areas. What is
?
and
divide
nonsquare rectangles. The sum of the areas of the two squares is
Find
respectively. The line through
into two squares and two
of the area of square
(AIME 2013)
25. The isosceles right triangle
has right angle at and area
intersect
at and . What is the area of
? (AMC 2015)
. The rays trisecting