- Chapter 1: Foundations for Geometry
- Chapter 1-1: Understanding Points, Lines, and Planes
- Chapter 1-2: Measuring and Constructing Segments
- Chapter 1-3: Measuring and Constructing Angles
- Chapter 1-4: Pairs of Angles
- Chapter 1-5: Using Formulas in Geometry
- Chapter 1-6: Midpoint and Distance in the Coordinate Plane
- Chapter 1-7: Transformations in the Coordinate Plane
- Chapter 10: Spatial Reasoning
- Chapter 10-1: Solid Geometry
- Chapter 10-2: Representations of Three-Dimensional Figures
- Chapter 10-3: Formulas in Three Dimensions
- Chapter 10-4: Surface Area of Prisms and Cylinders
- Chapter 10-5: Surface Area of Pyramids and Cones
- Chapter 10-6: Volume of Prisms and Cylinders
- Chapter 10-7: Volume of Pyramids and Cones
- Chapter 10-8: Spheres
- Chapter 11: Circles
- Chapter 11-1: Lines That Intersect Circles
- Chapter 11-2: Arcs and Chords
- Chapter 11-3: Sector Area and Arc Length
- Chapter 11-4: Inscribed Angles
- Chapter 11-5: Angle Relationships in Circles
- Chapter 11-6: Segment Relationships in Circles
- Chapter 11-7: Circles in the Coordinate Plane
- Chapter 12: Extending Transformational Geometry
- Chapter 12-1: Reflections
- Chapter 12-2: Translations
- Chapter 12-3: Rotations
- Chapter 12-4: Compositions of Transformations
- Chapter 12-5: Symmetry
- Chapter 12-6: Tessellations
- Chapter 12-7: Dilations
- Chapter 2: Geometric Reasoning
- Chapter 2-1: Using Inductive Reasoning to Make Conjectures
- Chapter 2-2: Conditional Statements
- Chapter 2-3: Using Deductive Reasoning to Verify Conjectures
- Chapter 2-4: Biconditional Statements and Definitions
- Chapter 2-5: Algebraic Proof
- Chapter 2-6: Geometric Proof
- Chapter 2-7: Flowchart and Paragraph Proofs
- Chapter 3: Parallel and Perpendicular Lines
- Chapter 3-1: Lines and Angles
- Chapter 3-2: Angles Formed by Parallel Lines and Transversals
- Chapter 3-3: Proving Lines Parallel
- Chapter 3-4: Perpendicular Lines
- Chapter 3-5: Slopes of Lines
- Chapter 3-6: Lines in the Coordinate Plane
- Chapter 4: Triangle Congruence
- Chapter 4-1: Classifying Triangles
- Chapter 4-2: Angle Relationships in Triangles
- Chapter 4-3: Congruent Triangles
- Chapter 4-4: Triangle Congruence: SSS and SAS
- Chapter 4-5: Triangle Congruence: ASA, AAS, and HL
- Chapter 4-6: Triangle Congruence: CPCTC
- Chapter 4-7: Introduction to Coordinate Proof
- Chapter 4-8: Isosceles and Equilateral Triangles
- Chapter 5: Properties and Attributes of Triangles
- Chapter 5-1: Perpendicular and Angle Bisectors
- Chapter 5-2: Bisectors of Triangles
- Chapter 5-3: Medians and Altitudes of Triangles
- Chapter 5-4: The Triangle Midsegment Theorem
- Chapter 5-5: Indirect Proof and Inequalities in One Triangle
- Chapter 5-6: Inequalities in Two Triangles
- Chapter 5-7: The Pythagorean Theorem
- Chapter 5-8: Applying Special Right Triangles
- Chapter 6: Polygons and Quadrilaterals
- Chapter 6-1: Properties and Attributes of Polygons
- Chapter 6-2: Properties of Parallelograms
- Chapter 6-3: Conditions for Parallelograms
- Chapter 6-4: Properties of Special Parallelograms
- Chapter 6-5: Conditions for Special Parallelograms
- Chapter 6-6: Properties of Kites and Trapezoids
- Chapter 7: Similarity
- Chapter 7-1: Ratio and Proportion
- Chapter 7-2: Ratios in Similar Polygons
- Chapter 7-3: Triangle Similarity: AA, SSS, and SAS
- Chapter 7-4: Applying Properties of Similar Triangles
- Chapter 7-5: Using Proportional Relationships
- Chapter 7-6: Dilations and Similarity in the Coordinate Plane
- Chapter 8: Right Triangles and Trigonometry
- Chapter 8-1: Similarity in Right Triangles
- Chapter 8-2: Trigonometric Ratios
- Chapter 8-3: Solving Right Triangles
- Chapter 8-4: Angles of Elevation and Depression
- Chapter 8-5: Law of Sines and Law of Cosines
- Chapter 8-6: Vectors
- Chapter 9: Extending Perimeter, Circumference, and Area
- Chapter 9-1: Developing Formulas for Triangles and Quadrilaterals
- Chapter 9-2: Developing Formulas for Circles and Regular Polygons
- Chapter 9-3: Composite Figures
- Chapter 9-4: Perimeter and Area in the Coordinate Plane
- Chapter 9-5: Effects of Changing Dimensions Proportionally
- Chapter 9-6: Geometric Probability
Geometry 1st Edition - Solutions by Chapter
Full solutions for Geometry | 1st Edition
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].
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.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
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).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Outer product uv T
= column times row = rank one matrix.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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 picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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