 Chapter 0: Geometric Art
 Chapter 0.1: Geometry in Nature and in Art
 Chapter 0.2: Line Designs
 Chapter 0.3: Circle Designs
 Chapter 0.4: Op Art
 Chapter 0.5: Knot Designs
 Chapter 0.6: Islamic Tile Designs
 Chapter 1: Introducing Geometry
 Chapter 1.1: Building Blocks of Geometry
 Chapter 1.2: Poolroom Math
 Chapter 1.3: Whats a Widget?
 Chapter 1.4: Polygons
 Chapter 1.5: Triangles
 Chapter 1.6: Special Quadrilaterals
 Chapter 1.7: Circles
 Chapter 1.8: Space Geometry
 Chapter 1.9: A Picture Is Worth a Thousand Words
 Chapter 10: Volume
 Chapter 10.1: The Geometry of Solids
 Chapter 10.2: Volume of Prisms and Cylinders
 Chapter 10.3: Volume of Pyramids and Cones
 Chapter 10.4: Volume Problems
 Chapter 10.5: Displacement and Density
 Chapter 10.6: Volume of a Sphere
 Chapter 10.7: Surface Area of a Sphere
 Chapter 11: Similarity
 Chapter 11.1: Proportion and Reasoning
 Chapter 11.2: Similar Triangles
 Chapter 11.3: Indirect Measurement with Similar Triangles
 Chapter 11.4: Corresponding Parts of Similar Triangles
 Chapter 11.5: Proportions with Area
 Chapter 11.6: Proportions with Volume
 Chapter 11.7: Proportional Segments Between Parallel Lines
 Chapter 12: Transforming Functions
 Chapter 12.1: Trigonometric Ratios
 Chapter 12.2: Problem Solving with Right Triangles
 Chapter 12.3: The Law of Sines
 Chapter 12.4: The Law of Cosines
 Chapter 12.5: Problem Solving with Trigonometry
 Chapter 13: Geometry as a Mathematical System
 Chapter 13.1: The Premises of Geometry
 Chapter 13.2: Planning a Geometry Proof
 Chapter 13.3: Triangle Proofs
 Chapter 13.4: Quadrilateral Proofs
 Chapter 13.5: Indirect Proof
 Chapter 13.6: Circle Proofs
 Chapter 13.7: Similarity Proofs
 Chapter 13.8: Coordinate Proof
 Chapter 2: Reasoning in Geometry
 Chapter 2.1: Inductive Reasoning
 Chapter 2.2: Finding the nth Term
 Chapter 2.3: Mathematical Modeling
 Chapter 2.4: Deductive Reasoning
 Chapter 2.5: Angle Relationships
 Chapter 2.6: Special Angles on Parallel Lines
 Chapter 3: 3 Using Tools of Geometry
 Chapter 3.1: Duplicating Segments and Angles
 Chapter 3.2: Constructing Perpendicular Bisectors
 Chapter 3.3: Constructing Perpendiculars to a Line
 Chapter 3.4: Constructing Angle Bisectors
 Chapter 3.5: Constructing Parallel Lines
 Chapter 3.6: Construction Problems
 Chapter 3.7: Constructing Points of Concurrency
 Chapter 3.8: The Centroid
 Chapter 4: Discovering and Proving Triangle Properties
 Chapter 4.1: Triangle Sum Conjecture
 Chapter 4.2: Properties of Isosceles Triangles
 Chapter 4.3: Triangle Inequalities
 Chapter 4.4: Are There Congruence Shortcuts?
 Chapter 4.5: Are There Other Congruence Shortcuts?
 Chapter 4.6: Corresponding Parts of Congruent Triangles
 Chapter 4.7: Flowchart Thinking
 Chapter 4.8: Proving Special Triangle Conjectures
 Chapter 5: Discovering and Proving Polygon Properties
 Chapter 5.1: Polygon Sum Conjecture
 Chapter 5.2: Exterior Angles of a Polygon
 Chapter 5.3: Kite and Trapezoid Properties
 Chapter 5.4: Properties of Midsegments
 Chapter 5.5: Properties of Parallelograms
 Chapter 5.6: Properties of Special Parallelograms
 Chapter 5.7: Proving Quadrilateral Properties
 Chapter 6: Discovering and Proving Circle Properties
 Chapter 6.1: Tangent Properties
 Chapter 6.2: Chord Properties
 Chapter 6.3: Arcs and Angles
 Chapter 6.4: Proving Circle Conjectures
 Chapter 6.5: The Circumference/ Diameter Ratio
 Chapter 6.6: Around the World
 Chapter 6.7: Arc Length
 Chapter 7.1: Transformations and Symmetry
 Chapter 7.2: Properties of Isometries
 Chapter 7.3: Compositions of Transformations
 Chapter 7.4: Tessellations with Regular Polygons
 Chapter 7.5: Tessellations with Nonregular Polygons
 Chapter 7.6: Tessellations Using Only Translations
 Chapter 7.7: Tessellations That Use Rotations
 Chapter 7.8: Tessellations That Use Glide Reflections
 Chapter 8: Area
 Chapter 8.1: Areas of Rectangles and Parallelograms
 Chapter 8.2: Areas of Triangles, Trapezoids, and Kites
 Chapter 8.3: Area Problems
 Chapter 8.4: Areas of Regular Polygons
 Chapter 8.5: Areas of Circles
 Chapter 8.6: Any Way You Slice It
 Chapter 8.7: Surface Area
 Chapter 9: Circles and the Pythagorean Theorem
 Chapter 9.1: The Theorem of Pythagoras
 Chapter 9.2: The Converse of the Pythagorean Theorem
 Chapter 9.3: Radical Expressions
 Chapter 9.4: Story Problems
 Chapter 9.5: Distance in Coordinate Geometry
 Chapter 9.6: Circles and the Pythagorean Theorem
Discovering Geometry: An Investigative Approach 4th Edition  Solutions by Chapter
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Discovering Geometry: An Investigative Approach  4th Edition  Solutions by Chapter
Get Full SolutionsThis expansive textbook survival guide covers the following chapters: 112. Since problems from 112 chapters in Discovering Geometry: An Investigative Approach have been answered, more than 8594 students have viewed full stepbystep answer. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. The full stepbystep solution to problem in Discovering Geometry: An Investigative Approach were answered by , our top Math solution expert on 03/13/18, 07:09PM.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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