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Transcript
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: 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: 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. Right angles Acute angles Obtuse angles Complementary angles Supplementary angles Now let’s look at the role of angles in lines Alternate-interior angles Corresponding angles Exterior angles Let’s look at some of the properties of angles in regular polygons 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 o 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: o Area of a regular hexagon: o # 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 Acute triangle Right triangle Obtuse triangle Equilateral triangle Isosceles triangle Legs 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 Medians Centroid Angle bisectors Incenter Perpendicular bisector Circumcenter Altitudes Orthocenter There are also important formulas associated with triangles o o The Triangle Inequality Pythagorean theorem Degenerate Pythagorean Triples We can use angles and sides to culminate another important concept: Congruent and Similar Triangles. 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. Inscribed angles Central angles Radius Center Diameter 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: o o o o o 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