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_____ Geometry B — Semester Overview General Formulas • Distance Formula: Given P=(xi,yi) and Q • Slope Formula: Given P=(xi,yj) and Q • = = (x2,y2), distance is d (x2,y2), slope is m = J(x —x 1 +(y 2 2 1 ) 7 —y,) = xl —x 2 Midpoint Formula: Given P=(xj,yi) and 2 ±X Q (x2,y2), midpoint is = Y ±Y 2 Polygons Poly2on Characteristics: • • • • • • • • Closed flat geometric figures No side is curved Consecutive sides intersect oniy at endpoints Nonconsecutive sides don’t intersect Each vertex must belong to exactly 2 sides Consecutive sides must be non-collinear Each segment of polygon is called side and each endpoint of side is called vertex. The number of sides is always equal to the number of vertices. Identifying Po1yons: • • Name each polygon by the number of its sides. Identify if convex/concave and regular/irregular. o Regular convex polygon where all sides/angles are congruent — Poiygon Formuias: • Sum Si of measures of angles of polygon with n sides: S • If I exterior angle is taken at each vertex, sum S of measures of exterior angles of polygon: Se • Number d of diagonals that can b drawn in polygon of n sides: d • Measure E of each exterior angle of equiangular polygon of n sides: E • Sum of measure of 3 angles of triangle is 180 • Exterior Angle of Polygon: adjacent & supplementary to interior angle of polygon. Measure equals sum of measures of remote interior angles. = (n-2)180 = 360 n(n —3) RevA Geometry B — Semester Overview Proportions/Similarity • • • • • Perimeter Ratio Theorem: The ratio of the perimeters of 2 similar polygons equals the ratio of any pair of corresponding sides. AA’-. Theorem: if 2 Ls of I A are to 2 /s of other A. then As are SSS Theorem: if measures of corresponding sides of 2 As are proportional then As are SAS- Theorem: if measures of 2 sides of A are proportional to measures o12 corresponding sides of then As are another A and included angles are Given the diagram to the right, the following apply A Bz’ E ifBEJCD.=-(B ED C”z’ D ir-= then BE Ci) ED (B if B and E are midpoints of AC and • CD and BE CD = Three Parallel Lines Corollary: if 3 or more parallel lines are intersected by 2 transversals, the A parallel lines divide the transversals proportionally. If AB H CD H EF, BE (E 4E BD AE 4( = = DF DF CE DE BE CE angle of triangle. it divides opposite side into segments an Angle Bisector Theorem: if a ray bisects that are proportional to adjacent sides. then • then BE — —. — ——, If AC bisects ZBAD then BC[) • Proportions: 2 or more ratios are equal o a is l term a = c uid or a:b = c:d term a b is 2 rd term oc is 3 th term ad is 4 -— — th 4 terms o Extremes: 1 & terms a Means: 2 & • Mean ProportionallGeometric Means: (means are the same i.e. aix o Example: find geometric means of I and 16 • 1/x = x/l6 (Set the proportions) • I * 16 = xx (Cross multiply) 16 = x (Simplify) • \u16x2 (Solvetbrx) • +4x • Given 2 squares with sides x and y. ratios of o Perimeters is x:v 2 o Area is x:y 2 ‘ = x/r) RevA __ ____________ Geometry B — Semester Overview Right Triangles/Trigonometry Finding missing lengths of right triangle When an altitude is drawn from the vertex of a right angle of the right triangle to its hypotenuse. then the Ibllowing apply: A • Altitude geometric means between c 1 measures of the 2 segments of the hypotenuse b o alt= 2 ,Jci*c C2 • Sides geometric means of hypotenuse and part of hypotenuse adjacent to the side. o a+c,) o C a B hc(c+c,) Classify Type of triangle: c is length of longest side of triangle o o o a 2 a 2 a ± + + h>c 2 2 triangle is acute 2 triangle is right 2=c b <c triangle is obtuse 22 b 2 Pythagorean Theorem: a o + h 2=c 2 Triples: 3-4-5. 5-12-13, 7-24-25, 8-15-17. 9-40-41 Special Right Triangles l Solving Right Triangles Trig Functions - • SOl I-CAB-bA • Sin = • • = LI A CosA Tan A h’poic’nuse c ad/uceni h hvpownuse c opposite adjacent a = C h 3 b A RevA Geometry B — Semester Overview Circles Formulas • • • • Radius = /2 Diameter Circumference of Circle: dir or 22tr 2 Area of Circle = irr 2 2 + (y-k) 2=r Equation of a circle: (x-h) Key Concepts • • • • . All radii are congruent Vertex on Center: angle = intercepted arc 2 (intercepted arc) Vertex on Circle: angle = V 2 (arc 1 + arc2) = average of bases Vertex inside circle: angle = V Vertex outside circle: angle = V2 (arc 1 arc2) difference of arcs — Finding the angle measure: Tangent-Secant Chord-Chord Tangent-Tangent Secant-Secant Inscribed Angle c (c—b) 2 a= V Finding the segment measure: Chord-Chord a = ½ a c rr ½ Secant-Secant Tangent-Secant 14 x I6 32 32(x) 16(12) I 2 142 = .1: 7(7 + x) 4 /1 AB(AC) = AD(AF) RevA (c -h) Geometry B — Semester Overview Area • If 2 closed regions intersect only along a common boundary, then the area of their union is equal to the sum of their individual areas. (This means break it up into pieces, find area of each piece and then add areas together.) Formulas: Formula Notes r Circle Kite 1 A dd 212 Parallelogram A = bh Rectangle A = 1w or A Rhombus Square Trapezoid is the radius d’s are the lengths of the diagonals of the kite A h is the perpendicular height to the base = bh or A 2 Length x width or - base x height bh 2 As A h is the altitude 1 = —h(b +b 1 ) 2 2 and the bs are the bases. M is the median of the Trapezoid Or A=Mh Triangle A Equilateral Triangle = — s is the side of the triangle s2T5 A= TriangleHero’s h is the altitude to the base !hh 2 4 AJ—s7)?c) Formula (a+b+c) 2 Use when there is no right angle or altitude Regular Polygon Sector of Circle A I P is the perimeter of polygon a & a is the apothem N A=—izr 360 N is the degree of the arc , 5 RevA Geometry B — Semester Overview Surface Area & Volume • • • • Volume: area of base x height Lateral Surface Area: sum of areas of lateral faces Surface Area: lateral surface area ± area of bases Triangular Prism: remember the base is the triangle not the rectangle. Note: The following apply to the k)rmulas below: o Ph Perimeter of the base o Ab Area of the base /) o B = area of the base a = slant height Sphere • • • = 4iti • 4 Vsphere = 3 Radii of 2 spheres with ratio x:y o Ratio of volumes: 3 :y x o Ratio of surface areas: 2 :y x = • SAcone 011 LA = Bh Vtiem,spiiere + Aconchase itr (‘ = r A Lzpyra rnij + 2 mr Note: use slant height for surface area and height for volume, SArcet = 2(lh) + Vreet = • = LArect — 2 • SApyrarnid • Vpyraniid • Note: use slant height for surface area and height for volume. = LApyranhid + Apr,c Ahh Cube = + 2(hw) + Bh lwh 2(l+w) * h 2*A hasc 2(l+w)h 2(1w) + 2(1w) or Triangular Prism • • • = 3 = Rectangle • LArect = Ph * h • A base = l*w • 4 (2tr 1) = Aeoiiebase • 2 37tr 2 SAheniispherc Pyramid Ch = • • C = circumference of the circle h = height hemisphere SAsphere Cone • LA 11 • a a • • • LAb SAb = 4(s ) 2 6(s) Bh lwh = = 3 s Cylinder LAtriaiig Pi, * h A(riangbase = see area section for options SAtriang LAiriang + 2 Atrianghasc Viriajig = Bh • • • 22tr • 6 LAcylinder = Ch = 2itrh Acslinderhase = itr SAcylinder LAcyiincer ± 2 Acylindcrbasc Veylinder = Bh = RevA 2rrh +