 6.4.1: Developing Proof In Exercises 14, the four conjectures are conseque...
 6.4.2: Developing Proof In Exercises 14, the four conjectures are conseque...
 6.4.3: Developing Proof In Exercises 14, the four conjectures are conseque...
 6.4.4: Developing Proof In Exercises 14, the four conjectures are conseque...
 6.4.5: Developing Proof For Exercises 57, determine whether each conjectur...
 6.4.6: Developing Proof For Exercises 57, determine whether each conjectur...
 6.4.7: Developing Proof For Exercises 57, determine whether each conjectur...
 6.4.8: MiniInvestigation Use what you know about isosceles triangles and ...
 6.4.9: Developing Proof Given circle O with chord and tangent AB BC in the...
 6.4.10: For each of the statements below, choose the letter for the word th...
 6.4.11: Match each term in Exercises 1119 with one of the figures AN. Minor...
 6.4.12: Match each term in Exercises 1119 with one of the figures AN. Major...
 6.4.13: Match each term in Exercises 1119 with one of the figures AN. Semic...
 6.4.14: Match each term in Exercises 1119 with one of the figures AN. Centr...
 6.4.15: Match each term in Exercises 1119 with one of the figures AN. Inscr...
 6.4.16: Match each term in Exercises 1119 with one of the figures AN. Chord...
 6.4.17: Match each term in Exercises 1119 with one of the figures AN. Secan...
 6.4.18: Match each term in Exercises 1119 with one of the figures AN. Tange...
 6.4.19: Match each term in Exercises 1119 with one of the figures AN. Inscr...
 6.4.20: Developing Proof Explain why a and b are complementary.
 6.4.21: What is the probability of randomly selecting three collinear point...
 6.4.22: Developing Proof Use the diagram at right and the flowchart below t...
Solutions for Chapter 6.4: Proving Circle Conjectures
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 6.4: Proving Circle Conjectures
Get Full SolutionsSince 22 problems in chapter 6.4: Proving Circle Conjectures have been answered, more than 24860 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.4: Proving Circle Conjectures includes 22 full stepbystep solutions. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824.

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!

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

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.

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).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

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!

Skewsymmetric 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).

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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