 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 20101 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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.