Geometry Chapters 6-11 Flashcards

Chapter 6: Basic Transformations and Similarity

• Similar polygons have congruent corresponding angles and proportional corresponding sides.
• Reflection creates a mirror image across a line or axis.
• Rotation involves turning a figure around a specific point.
• Translation refers to sliding a figure in any direction without altering its shape.

Chapter 7: Triangle Similarity

• Triangles can be proven similar using:
  • AA (Angle-Angle) postulate
  • SSS (Side-Side-Side) similarity
  • SAS (Side-Angle-Side) similarity.
• The 45-45-90 theorem relates the lengths of the legs to the hypotenuse: hypotenuse = leg * √2.
• In a 30-60-90 triangle, the relationships are:
  • Hypotenuse = 2 × shorter leg
  • Longer leg = shorter leg × √3.

Chapter 8: Angles in Polygons

• The sum of interior angles in a quadrilateral is 360 degrees.
• To find the angle sum of any polygon, use the formula: (sides - 2) * 180.
• According to the Polygon Exterior Angles Theorem, the sum of exterior angles of any polygon equals 360 degrees.

Quadrilaterals Characteristics

• A parallelogram has opposite sides that are equal and parallel, with bisecting diagonals and congruent opposite angles.
• A rhombus is a type of parallelogram with four equal sides and perpendicular diagonals.
• A rectangle is a parallelogram with four right angles.
• A square is a special parallelogram that has both four right angles and four congruent sides.
• A trapezoid features one pair of parallel sides known as bases.
• An isosceles trapezoid has congruent base angles and equal diagonals.

Chapter 9 and 10: Circles and their Properties

• Tangent circles touch at exactly one point.
• Concentric circles share the same center but have different radii.
• The standard equation of a circle with center (h, k) and radius r is given by: (x - h)² + (y - k)² = r².

Transformations and Vectors

• A translation can be described by the notation: (x,y) → (x+a, y+b).
• The component form of a vector consists of its horizontal and vertical movements.
• Reflection in the x-axis transforms a point (x, y) to (x, -y).
• Reflection in the y-axis transforms a point (x, y) to (-x, y).
• Reflection in the line y = x changes a point (b, a) to (b, a).
• Rotation transformations involve:
  • 90 degrees: (-b, a)
  • 180 degrees: (-a, -b)
  • 270 degrees: (b, -a).

Chapter 11: Measurement of Circles and Solids

• The circumference of a circle can be calculated using either 2πr or πd.
• Arc length determination uses the ratio: Length/2πr = Measure/360.
• The area of a circle can be found using the formula: πr².
• The area of a sector of a circle follows: Area/πr² = Measure/360.

Areas and Volumes of Solids

• The area of a right cone includes the base area (πr²) plus the lateral area (πrl).
• The area of a cylinder combines the base areas and lateral surface area: 2πr² + 2πrh.
• The surface area of a sphere is calculated as 4πr².
• The area of a trapezoid is given by: 1/2h(b1+b2).
• Volume calculations are characterized by:
  • Cylinder: πr²h
  • Cone: 1/3πr²h
  • Pyramid: 1/3Bh
  • Sphere: 4/3πr³.

Euler's Theorem and Circle Terminology

• Euler's Theorem states the relation: V + F = E + 2, where V is vertices, F is faces, and E is edges.
• A chord is a segment with endpoints on the circle.
• A secant is a line that intersects a circle at two points.
• A tangent touches a circle at exactly one point.
• A central angle has its vertex at the circle's center.
• Minor and major arcs are segments of a circle, defined by their angles.
• Adjacent arcs are arcs that intersect at a single point.
• An inscribed angle is formed by two chords from the circle.

Coordinate Geometry

• The distance formula calculates the distance between two points in the coordinate plane.
• The midpoint formula determines the midpoint between two endpoints on a line segment.
• The slope is defined as the ratio of the rise over the run between two points on a line.