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Transcript
1 2 Formula (given): Volume of a Pyramid V = 1/3 BH What does B represent? Formula: Area of a Trapezoid 3 4 Centroid 5 Midsegment of a triangle 6 Point Slope Form of Linear Equation *can be used to write equation of a line Slope formula 7 8 Slope-Intercept Form of linear equation 9 Quadrilateral Family Tree Midsegment of a Trapezoid 10 Standard Form Of Linear Equation Area of the base. B = triangle ½ bh B = Rectangle lw A = ½ h (b1 + b2) Where the medians of a triangle intersect. Each median is divided into a 2:1 ratio. The segment closest to vertex is twice as long! π= y - y = m(x - x ) 1 1 y 1 = y-coordinate x 1 = x-coordinate m = slope Rise Run π¦2 βπ¦1 π₯2 βπ₯1 Ξy Ξx y = mx + b m = slope b = y-intercept Ax + By = C where A,B, and C are real numbers Substitute 0 in for x to find the y-intercept Substitute 0 in for y to find the x-intercept 11 12 Perpendicular Lines 13 Parallel Lines 14 Parallelogram Properties 15 Rectangle Properties 16 Square Properties 17 Rhombus Properties 18 Kite Properties 19 Trapezoid Properties 20 Isosceles Trapezoid Quadrilaterals Same Slopes Negative Reciprocal Slopes (Opposite Signs and Flipped) 2 β3 3 2 Example: and At least 1 pair of parallel sides 21 22 Constructing: Perpendicular Bisector Midpoint 23 Constructing: Angle Bisector 24 Constructing: Equilateral Triangle 60° angle 25 Constructing: Parallel Lines 26 Constructing: Perpendicular from a Point on the line 27 Constructing: Perpendicular from a point NOT on the line 28 Reflection in x-axis (x, y) 29 Reflection in line y = x (x, y) Reflection in y-axis (x, y) 30 Reflection in line y = -x (x, y) (- x, y) (x, - y) (-y, -x) (y, x) 31 32 Point reflection through the Origin (same as Rotation 180° CC) (x, y) 33 R90 34 Isometry (same as RIGID MOTION) Preservesβ¦ R270 35 36 Opposite isometry What transformations are rigid motions (isometries)? 37 38 Direct Isometry 39 Composition of Transformations 40 Orthocenter Incenter (-x, -y) (-y, x) (y, -x) orientation is not preserved - the order of the lettering is reversed Line Reflections Reflections, rotations, translations, and glide reflections are all examples of rigid motions. (NOT DILATIONS) orientation is preserved - the order of the lettering in the figure and the image are the same Rotations, translations, POINT reflections Intersection of Angle Bisectorss Intersection of Altitudes 41 42 Circumcenter 43 Isosceles Triangle 44 Special Right Triangle 30°-60°-90° 45 Special Right Triangle 45°-45°-90° 46 How do you determine the longest side of a triangle? 47 Triangle Exterior Angle Theorem 48 Altitude to Hypotenuse 49 Distance Formula 50 Slope Formula Midpoint Formula Intersection of Perpendicular Bisectors 45°-45°-90° x x 30°-60°-90° x xβ3 2x Hidden in Equilateral Triangle with altitude. xβ 2 Hidden in Square w/diagonal If side opposite 90° is a whole #, ÷ β2 then rationalize to get side opposite 45°. Exterior angle = The sum of non-adjacent interior angles If side opposite 60° is a whole #, ÷ β3 then rationalize to get side opposite 30°. Opposite the largest angle ππ΅ πΏ πΏ =π» π π΄ =π π΄ 51 52 Cofunctions 53 Sine 54 Cosine 55 Tangent 56 Pythagorean Theorem 57 When do you use Trig? 58 When do you use the Pythagorean Theorem? 59 How to determine possible length of 3rd side of a triangle. 60 Area of a Regular Polygon Apothem Sin < = πππ π»π¦π If trying to find angle use sin-1 tan < = πππ cos < = πππ If trying to find angle use tan-1 Use when angle is involved in RIGHT triangle πΊ Sine and Cosine They are = when angles are complementary. ο· Sin (10°) = cos (80°) ο· Find value of x: sin (x) = cos (x + 5) Add angles and set to 90. πΆ π¨ πΆ πͺ π» π― π― π¨ Given side, angleβ¦Find other side Given 2 sidesβ¦find angle Given 2 sides of a triangle Subtract them Add them 3rd side must be between those lengths Ex. 5 and 7 3rd side must be between 2 and 12 (not 2 or 12) 1 Area of reg poly= πππππππ‘ππ β ππππ‘βππ 2 πππ π»π¦π If trying to find angle use cos-1 Triples: 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17 Use when only sides of a right triangle are involved. Know 2 sides of a right triangle Asked to find the third side Segment drawn from the center of a regular polygon to the midpoint of a side (will also be β₯) 61 62 2 tangents to a circle (Segment/Angle Relationship) 63 2 chords in a circle (Segment/Angle Relationship) 64 2 secants to a circle (Segment relationship) 65 Angle formed Outside the Circle (angle relationship) 67 Tangent /Secant (Segment Relationship) 66 Equation of a circle (center at the origin) 68 Central Angle 69 Equation of a Circle (center shifted) 70 Inscribed Angles Watch βOut Secant/Tangent Segments Angles ο· ο· tan = tan minor arc is supplementary to angle formed by 2 tangents Pβ’P=Pβ’P Aβ’B=Cβ’D w β’ e = t2 (b+c) b = a2 we=we (a+b)β’b = (c+d)β’d Measure of angle = Measure of arc ο· opp signs of center! o ½ arc o Subtract from 180 Measure of angle = ½ arc 71 72 Which point of concurrency locates the point equidistant from all 3 vertices of a triangle? Points of Concurrency 73 74 Area of a KITE/Rhombus/Square 75 Length of an Arc (degrees) 76 Area of a Sector of a Circle 77 Length of an Arc (Radians) 78 Similar Triangles 4 proportions GOOD 1 proportion NOT GOOD! Converting between degrees and radians 79 Similar Triangle Proofs (last 3 steps) 80 Inscribed Angle that intercepts the diameter All Of My Children Are Bringing In Peanut Butter Cookies Perpendicuar Bisectors- Circumcenter Altitudes-Orthocenter Medians- Centroid Angle Bisectors- Incenter Perpendicuar Bisectors- Circumcenter A = ½ d1 β’d2 Portion of the Circumference L= π° 360 β ππ S = ππ Portion of the Area L= π ππ ππππ‘πππ πππππ ππ π π΄π·πΌπ΄ππ GOOD ο· ο· ο· ο· π΄π· = π·π΅ π΄π· π΄π΅ π·π΅ π΄π΅ π΄π· π·πΈ π° 360 β ππ 2 NOT GOOD π΄πΈ πΈπΆ π΄πΈ PIECE/PIECE π΄π· π·πΈ = π·π΅ π΅πΆ = π΄πΆ PIECE/WHOLE πΈπΆ = π΄πΆ PIECE/WHOLE = π΄π΅ π΅πΆ WHOLE/WHOLE Is a right angle! Ξ ~ Ξ = ( )( ) = ( )( ) AA CSSTP In a proportion, the product of the means = product of the extremes. 81 82 Cavalieriβs Principle 83 Dilation with Respect to the ORIGIN 84 Segment Perpendicular to a chord Dilation with Respect to a POINT 85 86 What does CSSTP stand for? What does CPCTC stand for? 87 What congruence criteria can be used to prove 2 triangles are congruent? 89 Dilating a LINE With respect to origin 88 How do you prove 2 triangles are Similar? 90 Angle of Elevation/Depression Multiply pre-image points by scale factor. DK, O (x, y) = (kx, ky) Bisects the chord Corresponding Sides of Similar Triangles are Proportional Two solids will have equal volumes if their bases have equal area and their altitudes (heights) are equal Point of dilation, image and pre-image must be collinear. Corresponding Parts of Congruent Triangles are Congruent AA Multiply y-intercept by scale factor SLOPE STAYS SAME! Ex. D2,O y= 2x β 4 Becomes y = 2x β 8 91 92 Why are all circles similar? Altitude of an isosceles triangle is also β¦ 93 Writing an equation of perpendicular bisector 94 95 96 How to construct a SQUARE INSCRIBED IN A CIRCLE Partitioning a Directed Segment How to construct a Equilateral Triangle / Hexagon INSCRIBED IN A CIRCLE 97 98 Rotational Symmetry 99 Point Symmetry 100 Coordinate Geometry Proofs Completing the Square for a CIRCLE x2 + y2 + cx + dy = e Parallelogram Rectangle Rhombus Square Trapezoid Isosceles Trapezoid Bisects Vertex Angle Bisects the BASE Composition of a Translation and Dilation. 1st Find Midpoint 2nd Find Slope 3rd find Negative Reciprocal of slope. Then use y β y1 = m(x β x1) with your midpoint and NR slope Divide the slope (rise and run) Example 2:3 ratio, divide rise and run by 5. Then count new slope to create partition Looks the SAME after a rotation of 180 (turn upside down) Parallelogram: 4 SLOPES (Opp. Sides ||) Rectangle: 4 SLOPES (Opp. Sides ||, 1 pair consecutive sides β₯) Rhombus: 6 SLOPES (Opp. Sides ||, diagonals β₯) Square: 6 SLOPES (Opp. Sides ||, 1 pair consecutive sides β₯, diagonals β₯) Trapezoid: 2 SLOPES (1 pair of || sides) Isosceles Trapezoid: 2 SLOPES, 2 DIST (1 pair of || sides, non || sides congruent) Measure radius, mark off radius (creating 60° central angle) Looks the SAME after a rotation Maps onto itself π 2 ( ) , πππ π‘π πππ‘β π ππππ 2