Triangles: Geometry Rules 1. Types of Triangles: By Sides: Scalene - no congruent sides Isosceles - 2 congruent sides Equilateral - 3 congruent sides By Angles: Acute - all acute angles Right - one right angle Obtuse - one obtuse angle Equiangular - 3 congruent angles - each are 60 2. Pythagorean Theorem for Right Triangles 3. Interior and Exterior Angles of Triangles 1 + 2 + 3 = 180 1 + 4 = 180 2 + 3 = 4 4. Isosceles Triangle Base angles are equal
5. 3 sides of a Triangle: If you add any 2 sides the result must be larger than the 3rd side. Possibilities for 3rd side when given 2 sides: Subtract the two sides < missing side < Add the two sides 6. Relationship between sides and angles in a triangle: The largest angle is across from the longest side. The smallest angle is across from the shortest side. The longest side is across from the largest angle. The smallest side is across from the smallest angle. 7. Side Splitter Theorem: If a line joining 2 sides of a is parallel to the 3rd side, then the following proportion can be made: 8. Midsegments: D and E are midpoints of AB and AC DE is parallel to BC and DE = 1 2 BC 9. Centroid: The point where all 3 medians of the triangle meet.
Special relationship of lengths within the median: The lengths of the median are in a 2:1 ratio 2x + x = length of median 10. Orthocenter - the point where all 3 altitudes of the triangle meet. 11. Incenter - the point where all the angle bisectors of the triangle meet. It is also the center of the circle that is inscribed in the triangle. 12. Circumcenter - the point where all the perpendicular bisectors of the sides of the triangle meet. It is also the center of the circle that is circumscribed around the triangle. Coordinate Geometry 13. Midpoint - Used to find the middle of a segment or location of the perpendicular bisector Also used to find the midpoints of the diagonals of parallelograms
14. Distance Formula - used to find the length of a segment Also used to prove that lengths are = (like the opposite sides of a parallelogram) 15. Slope Formula - used to find the slope of a line or segment Also used to prove lines parallel if slopes are = or used to prove lines perpendicular if slopes are negative reciprocals or used to show the existence of a right angle if the slopes are negative reciprocals 16. Equation of a line - Two forms Slope-Intercept Form: If you need to find the slope or y-intercept use: y = mx + b m = slope b = y-intercept Make sure that you have solved the equation for y. (y must be by itself on one side of the equation!) Point-Slope Form: If you need to write the equation of a line: y - y 1 = m(x - x 1 ) m = slope (x 1,y 1 ) is a point on the line Example: Write the equation of a line with slope = 3 and goes through the point (1, -4) y - y 1 = m(x - x 1 ) plug in values for m and x 1 and y 1 y - (-4) = 3(x - 1) Take care of double signs y + 4 = 3(x - 1) Solving for y at this point is optional. Example: Write the equation of a line that contains the points (-2,1) and (4,5) First find the slope - Rule #15 m = 5 1 4 2 = 4 6 = 2 3 y - y 1 = m(x - x 1 ) plug in values for m and x 1 and y 1 y - 1 = 2 3 (x - -2) Take care of double signs y - 1 = 2 3 (x + 2)
17. Circles: Equation of a circle: Example: Circle with center (2,-5), radius 3 Polygons 18. Regular Polygons: all angles = and all sides = 19. Sum of interior angles = 180(n - 2) where n is the number of sides 180(n 2) 20. Each interior angle = n for regular polygons 21. Sum of exterior angles = 360 360 22. Each exterior angle = n for regular polygons Angles, Arcs and Segments in Circles 23. 24. Inscribed Angle = 1 2 Arc 25. Angle formed by tangent/chord = 1 2 Arc
26. Angle formed by 2 chords = half the sum of the arcs 27. Angle formed by tangent/secant, or 2 secants, or 2 tangents = half the difference of arcs
28. Parallel chords intercept equal arcs: 29. A tangent and a radius are alway perpendicular at the point of tangency. 30. A radius that is perpendicular to a chord, bisects the chord and it s arcs. 31. Intersecting Chords Rule: a b = c d
32. Secant - Secant Rule: (whole secant) (external part) = (whole secant) (external part) b a = d c 33. Secant - Tangent Rule: (whole secant) (external part) = (tangent) 2 ( c + b) b = a 2 34. Hat Rule: Two Tangents are equal Quadrilaterals and Parallel Lines 35. Parallel lines and transversals Alternate interior angles two angles in the interior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate interior angles are non-adjacent and congruent.
Alternate exterior angles two angles in the exterior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate exterior angles are non-adjacent and congruent. Corresponding angles two angles, one in the interior and one in the exterior, that are on the same side of the transversal. Corresponding angles are non-adjacent and congruent. 36. Parallelograms: I have: - 2 sets of parallel sides - 2 sets of congruent sides - opposite angles congruent - consecutive angles supplementary - diagonals bisect each other - diagonals form 2 congruent triangles
37. Rectangle: same as parallelogram plus add 4 right angles and congruent diagonals 38. Rhombus: Same as parallelogram plus add 4 sides, diagonals are perpendicular and they bisect the opposite angles. 39. Square: All of the above properties (rules 36-38) 40. Transformational Geometry
Reflections: Glide Reflections: Translations: Rotations: 41. Composition of Transformations
42. Properties of Transformations: Orientation: Orientation refers to the arrangement of points, relative to one another, after a transformation has occurred. For example, the reference made to the direction traversed (clockwise or counterclockwise) when traveling around a geometric figure. Isometry: An isometry is a transformation of the plane that preserves length. A direct isometry preserves orientation or order - the letters on the diagram go in the same clockwise or counterclockwise direction on the figure and its image. An opposite isometry changes the order (such as clockwise changes to counterclockwise). 43. Quadratic - Linear Systems To solve by graphing: y= (x+3) 2-1 x + y = 2 Solve each equation for y and then put one equation in Y1 and the second in Y2 ZOOM 6 (to see in a standard window) If you can see all points of intersection then: 2nd TRACE 5 Press ENTER 3 times This will give you the (x,y) value of one intersection point. If there is another, 2nd TRACE 5 Press ENTER twice then arrow over to the other intersection point ENTER
44. Constructions Bisecting an Angle Perpendicular Bisector Parallel lines Perpendicular Line through a Point Equilateral Triangle 45. Locus Equidistant from a Point Equidistant from a line Equidistant from 2 points Equidistant from 2 Parallel Lines Equidistant from 2 intersecting lines
46. Similarity Polygons are similar if their corresponding (matching) angles are congruent (equal in measure) and the ratio of their corresponding sides are in proportion. Perimeters: Perimeter of ABC = AB Perimeter of DFE DE Areas: Area of ABC = AB 2 Area of DFE DE 2 47. Right Triangle Similarity
48. Logic Negations: In logic, a negation of a simple statement can usually be formed by placing the word "not into the original statement. The negation will always have the opposite truth value of the original statement. Under negation, what was TRUE, will become FALSE or what was FALSE, will become TRUE. Inverse: I Negate negate both sides of the conditonal Converse: Change Order switch the order of the conditonal Contrapositive: Change Order and Negate this is always logically equivalent to the original conditional 49. Truth Values Conjunction And True when both parts are true Disjunction Or False when both parts are false Conditional If/Then False when T F Biconditional If and only If True when both parts are the same. 50. Proving Triangles Congruent SSS SAS
ASA AAS HL 51. Proving Triangles Similar AA