- 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
Discovering Geometry: An Investigative Approach | 4th Edition - Solutions by ChapterGet Full Solutions
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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!
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].
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
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.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.