- 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
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
cond(A) = c(A) = IIAIlIIA-III = 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.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
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).
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).
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