 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 5394 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.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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