 5.7.1: Developing ProofIn Exercises 13, each conjecture has also been stat...
 5.7.2: Developing ProofIn Exercises 13, each conjecture has also been stat...
 5.7.3: Developing ProofIn Exercises 13, each conjecture has also been stat...
 5.7.4: Developing Proof Write a flowchart proof to demonstrate that quadri...
 5.7.5: The results of the proof in Exercise 4 can now be stated as a prove...
 5.7.6: Developing Proof For Exercises 69, prove the conjecture. 6. Conject...
 5.7.7: Developing Proof For Exercises 69, prove the conjecture. 6. Conject...
 5.7.8: Developing Proof For Exercises 69, prove the conjecture. 6. Conject...
 5.7.9: Developing Proof For Exercises 69, prove the conjecture. 6. Conject...
 5.7.10: You have discovered that triangles are rigid, but parallelograms ar...
 5.7.11: Find the measure of the acute angles in the 4pointed star in the I...
 5.7.12: A contractor tacked one end of a string to each vertical edge of a ...
 5.7.13: Find the equations of the lines containing the diagonals of rhombus...
 5.7.14: Find the point of intersection of the lines y = 1 and 3x 4y = 8.
 5.7.15: The last bus stops at the school some time between 4:45 and 5:00. W...
 5.7.16: The 3by9by12inch clear plastic sealed container shown is resti...
Solutions for Chapter 5.7: Proving Quadrilateral Properties
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 5.7: Proving Quadrilateral Properties
Get Full SolutionsChapter 5.7: Proving Quadrilateral Properties includes 16 full stepbystep solutions. Since 16 problems in chapter 5.7: Proving Quadrilateral Properties have been answered, more than 24826 students have viewed full stepbystep solutions from this chapter. 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. This expansive textbook survival guide covers the following chapters and their solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(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!

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).