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Transcript
Perimeter Circle (circumference) πΆ = 2ππ Square π = 4π Rectangle π = 2π + 2β B β area of base C β circumference π΄ = ππ ! π΄ = πβ Rectangle/Parallelogram π΄ = πβ ! r β radius π΄ = π! π! ! s β side V β volume Trapezoid π΄ = Regular polygon π΄ = ππ ! l β slant height P β perimeter ! ! d β diameter h β height ! Triangle Rhombus/Kite a β apothem b β base Area Circle A β area π! + π! β w β width ! ! Surface Area Lateral Total π = 4ππ ! Sphere Cone π = πππ π = πππ + ππ ! Cylinder π = 2ππβ π = 2ππβ + 2ππ ! Pyramid π = ππ π = ππ + π΅ Prism π = πβ π = πβ + 2π΅ ! ! Volume ! Sphere π = ππ ! Pyramid or Cone π = π΅β Prism or Cylinder π = π΅β ! ! ! ! ! Arcs and Sectors Degrees Radians r ! Arc Length π΄π΅ = Sector Area π΄ππ΅ = !"#° · 2ππ ! !"#° π΄π΅ = ππ O ΞΈ ! · ππ ! A π΄ππ΅ = π ! B ! Right Triangles π ! + π ! = π ! Pythagorean Theorem A Trigonometric Ratios (SOHCAHTOA) sin π΄ = cos π΄ = tan π΄ = !""!#$%& !"# csc π΄ = !"#$%&'()& !"#!$%&' !"# sec π΄ = !"#$%&'()& !""!#$%& !"# cot π΄ = !"#!$%&' !"# b !"#$%&'()& c !""!#$%& !"# !"#$%&'()& !"#!$%&' !"# !"#!$%&' !"# C !""!#$%& !"# a B Common Right Triangles 3-4-5 5-12-13 7-24-25 8-15-17 9-40-41 Special Triangles 60º 45º 2x π₯β2 x x x 30º 45º π₯β3 Non-Right Triangles Law of Sines !"# ! Law of Cosines π ! = π ! + π ! β 2ππ cos π΄ π ! = π ! + π ! β 2ππ cos π΅ π ! = π ! + π ! β 2ππ cos πΆ ! = !"# ! ! = A !"# ! ! c b B a C Coordinate Geometry Slope-intercept form of a linear equation π¦ = ππ₯ + π Point-slope form of a linear equation π¦ β π¦! = π(π₯ β π₯! ) Standard form of a linear equation π΄π₯ + π΅π¦ = πΆ, π€βπππ π΄ > 0 Slope of a line π= Midpoint of a line ( Distance formula π= Standard equation for a circle with center (π, π) (π₯ β π)! + (π¦ β π)! = π ! !! !!! !! !!! !! !!! !! !!! ! , ! ) (π₯! β π₯! )! + (π¦! β π¦! )! Miscellaneous Geometric Mean (π₯) ! ! ! = π¨π« π₯ ! = ππ ! Interior/Exterior Angles (regular polygon with n sides) Sum of interior angles = 180°(π β 2) Each interior angle = Sum of exterior angles = 360° Each exterior angle = !"#°(!!!) ! !"#° Similarity Ratios Length: ! ! Area: !! !! Volume: !! !! ! Commonly Used Theorems and Postulates Points, Lines, Planes, and Angles ο§ Segment Addition Postulate: If B is between A and C, then AB + BC = AC. ο§ Angle Addition Postulate: If point B is inside AOC, then m AOB + m BOC = m AOC. ο§ If two lines intersect, then they intersect at one point. ο§ If M is the midpoint of π΄π΅, then AM = ½AB and MB = ½AB. ο§ Angle Bisector Theorem: If π΅π bisects ABC, then m ABX = ½m ABC and m XBC = ½m ABC. ο§ Vertical Angles Theorem: Vertical angles are congruent. ο§ Right Angle Congruence Theorem: All right angles are congruent. ο§ Linear Pair Postulate: If two angles form a linear pair, then they are supplementary. ο§ If two angles are supplements or complements of congruent angles, then they are congruent. ο§ If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. ο§ If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Perpendicular and Parallel Lines ο§ If two lines form congruent adjacent angles, then the two lines are perpendicular. ο§ If two lines are perpendicular, then they form congruent adjacent angles. ο§ If two parallel lines are cut by a transversal, then/two lines cut by a transversal are parallel if: o the pairs of corresponding angles are congruent. o the pairs of alternate interior angles are congruent. o the consecutive interior angles are supplementary. o the alternate exterior angles are congruent. ο§ If two lines are parallel to the same line, then they are parallel to each other. ο§ In a plane, two lines perpendicular to the same line are parallel. Triangles ο§ The sum of the measures of the interior angles of a triangle is 180º. ο§ If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the two triangles are congruent. ο§ The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. ο§ The perpendicular bisectors of a triangleβs sides intersect at a point that is equidistant from the vertices of the triangle. ο§ The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. ο§ The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. ο§ Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. ο§ The bisector of the vertex angle of an isosceles triangle is perpendicular to the base of the triangle at its midpoint. ο§ The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. ο§ ο§ ο§ The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. Two triangles are congruent if: o the three sides of one are congruent to the three sides of the second (SSS). o two sides and the included angle of one are congruent to two sides and the included angle of the second (SAS). o two angles and the included sides of one are congruent to two angles and the included side of the second (ASA). o two angles and a non-included side of one are congruent to the corresponding parts of the second (AAS). o the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of a second right triangle (HL). Two triangles are similar if: o If two angles of one triangle are congruent to two angles of a second triangle (AA). o If an angle of one triangle is congruent to an angle in a second and the sides including those angles are proportional (SAS). o If the sides are in proportion (SSS). Quadrilaterals ο§ Area Addition Postulate: The area of a region is the sum of the areas of its non-overlapping parts. ο§ The sum of the measures of the interior angles of a quadrilateral is 360º. ο§ If a quadrilateral is a parallelogram: o its opposite sides are congruent. o its opposite angles are congruent. o its consecutive angles are supplementary. o its diagonals bisect each other. ο§ A quadrilateral is a parallelogram if: o both pairs of opposite sides are congruent. o both pairs of opposite angles are congruent. o an angle is supplementary to both of its consecutive angles. o the diagonals bisect each other. o one pair of opposite sides is congruent and parallel. ο§ A parallelogram is a rhombus if and only if: o its diagonals are perpendicular. o each diagonal bisects a pair of opposite angles. o two consecutive sides are congruent. ο§ A parallelogram is a rectangle if and only if: o the diagonals are congruent. o one angle is a right angle. ο§ Base angles of an isosceles trapezoid are congruent. ο§ A trapezoid is isosceles if its diagonals are congruent. ο§ The median of a trapezoid: o is parallel to the bases. o has a length equal to the average of the base lengths. ο§ ο§ ο§ If a quadrilateral is a kite: o its diagonals are perpendicular. o exactly one pair of opposite angles is congruent. A rhombus is both a parallelogram and a kite. A square is both a rectangle and a rhombus. Circles ο§ If a line is tangent to a circle, then the line is perpendicular to the radius drawn at the point of tangency. ο§ Tangents to a circle drawn from the same point are congruent. ο§ In a circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. ο§ In a circle or congruent circles: o congruent arcs have congruent chords. o congruent chords have congruent arcs. o chords equally distant from the center are congruent. ο§ A diameter or radius that is perpendicular to a chord bisects the chord and its arc. ο§ The measure of an inscribed angle is equal to half the measure of its intercepted arc. ο§ The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. ο§ The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle is equal to half the difference of the measure of the intercepted arcs. ο§ When two chords intersect inside a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the second. ο§ When two secant segments are drawn to a circle from an external point, the product of the length of one secant segment and the length of its external segment is equal to the product of the length of the other segment and the length of its external segment. ο§ When a secant segment and a tangent segment are drawn to a circle from and external point, the product of the length of the secant segment and the length of its external segment is equal to the square of the length of the tangent segment.
