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Transcript
Chapter 7 - Transformations (10) translations, reflections, rotations, line of symmetry, rotational symmetry, vectors, glide reflections Reflections on the x axis, the y axis & the line y = x Over x-axis (x, y) (x, -y) Over y-axis (x, y) (-x, y) Over the line y = x (x, y) (y, x) 90º counterclockwise, 270 clockwise about the origin (x, y) (-y, x) • If a point is rotated 90° clockwise, 270 counterclockwise about the origin, then (x , y) (y, -x) If a point is rotated 180° about the origin, then (x, y) (-x, -y) For a translation: (x,y) (x+a, y+b) or a,b (x+a, y+b) is the coordinate notation a,b> is the vector in component form Chapter 8 – Similarity (12) definition of similarity, proportions, similarity theorems, scale factor, dilations Similarity: sides proportional and angles congruent!!! RATIO: A comparison of two or more quantities in the same unit. written : a/b or a:b. Proportion: An equation which states two ratios are equal. a c b d ad bc a bb c d cb d d a c b d a bb c d cb d d a c b d Addition property of Proportions: a c b d ab cd b d a c b d a b c d Geometric Mean: a x x b a bb c d cb d d a c b d a bb c d cb d d a c b d Angle-Angle Similarity postulate: If two angles in one triangle are congruent to two angles in another triangle then the triangles are similar SSS Similarity Theorem: If the corresponding sides of two triangles are proportional, then the triangles are similar. SAS Similarity Theorem: If an angle in one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. Triangle Proportionality Theorem (Side - Splitter Theorem) Q T TU || QS if and only if RT RU TQ US R U S Parallel Lines & Proportions r If r || s and s || t, and l and m intersect at r, s, and t, then UW VX . WY XZ s t l U W Y V X Z m Angle Bisector Theorem If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. If CD bisects ACB , then AD CA . DB CB A D C B Dilation SCALE FACTOR: Ratio of any length on the image to the corresponding length on the original figure. image preimage(original ) To create a dilation about the origin: Multiply each coordinate by the scale factor Chapter 9 – Right Triangles & Trig (13) Pythagorean Theorem, Pythagorean triples, classifying triangles, altitude on hypotenuse theorem, geometric mean, 45-45-90 right triangle, 30-60-90 right triangle, trigonometric ratios. If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. ABC ~ CBD ~ ACD The length of the altitude is the geometric mean of the two segments. AD CD CD BD The length of each leg is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg BA CA CA AD Pythagorean Theorem Right Triangle: c 2 = a 2 + b 2 c a b A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c 2 = a 2 + b 2. Acute Triangle: c2 < a2 + b2 Obtuse Triangle: c 2 > a 2 + b 2 45 – 45 – 90 45 x√2 x 90 45 x 30 – 60 – 90 x 60 2x 90 x√3 30 Soh Cah Toa opposite Sin hypotenuse adjacent Cos hypotenuse opposite Tan adjacent Solving Right Triangle To find a side: ouse the given angle and decide sin, cos, or tan oset up an equation oSolve for unknown variable To find an angle: use inverse sin, cos, or tan: sinA = x, then sin-1(x) = A cosA = x, then cos-1(x) = m tanA = x, then tan-1(x) = m Solving Right Triangle Vector: Quantity with a magnitude and direction Magnitude (speed): The distance from the initial point to the terminal point AB x2 x1 y2 y1 2 2 Direction of a vector Determined by the angle it makes with a horizontal line X axis: east-west Y-axis: north-south Chapter 10 – Circles (14) definitions, radii-tangent theorems, measure of angles with relations to circles, equation of a circle A set of all points equidistant from a given point Circle: (called the center) Radius: Segment from the center to a point on the circle Diameter: Distance across the circle through the center Chord: Segment whose endpoints are points on the circle Secant: A line that intersects the circle in two points Tangent: A line that intersects the circle at exactly one point More Circles Internal Tangent: Tangent line intersects a line that joins External The centers of the circle Tangent: Does not intersect the segment that joins the center Circle Theorems Thm 10.1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency Thm 10.2 If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. Circle Theorems Thm 10.3 If two segments from the same exterior point are tangent to a circle, then they are congruent. x y x and y are tangent to the circle, so x = y . Angles and Arc Relationships Central angle = arc Inscribed angle = ½(arc) Angle inside circle = ½(sum) Angle outside circle = ½(difference) Theorem inscribed angles intercept the same arc, then the angles are congruent B A C D A B Since both angles intercept arc CD Theorems: A right triangle is inscribed in a circle the hypotenuse is a diameter of the circle. A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. wo y o 1800 x z 180 o o 0 x w y z Segment Relationships in Circles Two chords: (part)(part) = (part) (part) Two secants: (whole)(outside) = (whole)(outside) Secant and tan: tan2 = (whole)(outside) Standard Equation of a Circle (x - 2 h) + (y - 2 k) = 2 r (h, k) is the center r is the radius Note: To find the radius, you may need to use distance formulas d x2 x1 2 ( y2 y1 ) 2 Chapter 11 – Area of Polygons & Circles (12) area formulas for regular polygons, equilateral triangles, circles, arc length & area of sectors Polygon interior angles sum: 180(n – 2) Each polygon interior angle: 1 (n 2) 180 o n Polygon Exterior angles sum: 360o Each polygon interior angle: 360o/n Area of an Equilateral Triangle 1 2 A 3s 4 Area of a Regular Polygon: 1 A ans 2 360 central angle n Areas of similar polygons: a Sides: b Perimeter: a b 2 Area: a 2 b Circumference: The distance around a circle Formula to find circumference: c 2r Arc Length: The length of a portion of a circle Formula to find arc length : mAB Arc length of AB 2r 360 Area: The amount of space inside a circle Formula to find area: A r 2 Area of a Sector: The area of a piece of the circle (bounded by two radii) Formula to find the area of a sector : mAB 2 Area of AB r 360 Geometric Probability part Area shaded region whole Area entire region Chapter 12 – Surface Area & Volume (14) all surface area and volume formulas for prisms, cylinders, cones, pyramids, & spheres Eulers Theorem: The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula: F+V=E+2 To find edges: take ½ of the number of sides of each face Why? each side is shared by two polygons Surface Area of a Right Prism S = 2B +ph Surface Area of a Cylinder S 2r 2rh 2 Surface Area of a Pyramid S = B + ½Pl Surface Area of a Cone S r rl 2 Volume of Cube V 3 =s Volume of Prisms V = Bh Volume of Cylinder V= 2 Πr h Volume of Pyramids V = Bh 3 Volume of Cone V = Πr2h 3 Surface Area of a Sphere S 4 r 2 Volume of Sphere 4 3 V r 3 Theorem: If two similar solids have a scale factor of a:b, then the corresponding areas have a ratio of a2:b2, and corresponding volumes have a ratio of a3:b3.