 5.6.1: Developing Proof For Exercises 110, state whether each statement is...
 5.6.2: Developing Proof For Exercises 110, state whether each statement is...
 5.6.3: Developing Proof For Exercises 110, state whether each statement is...
 5.6.4: Developing Proof For Exercises 110, state whether each statement is...
 5.6.5: Developing Proof For Exercises 110, state whether each statement is...
 5.6.6: Developing Proof For Exercises 110, state whether each statement is...
 5.6.7: Developing Proof For Exercises 110, state whether each statement is...
 5.6.8: Developing Proof For Exercises 110, state whether each statement is...
 5.6.9: Developing Proof For Exercises 110, state whether each statement is...
 5.6.10: Developing Proof For Exercises 110, state whether each statement is...
 5.6.11: WREK is a rectangle. CR = 10 WE =
 5.6.12: PARL is a parallelogram. y =
 5.6.13: SQRE is a square. x = y =
 5.6.14: For Exercises 1416, use deductive reasoning to explain your answers.
 5.6.15: For Exercises 1416, use deductive reasoning to explain your answers.
 5.6.16: For Exercises 1416, use deductive reasoning to explain your answers.
 5.6.17: Construction Given the diagonal LV, construct square LOVE.
 5.6.18: Construction Given diagonal BKand B, construct rhombus BAKE.
 5.6.19: Construction Given side PS and diagonal PE , construct rectangle PIES.
 5.6.20: Developing Proof Write the converse of the Rectangle Diagonals Conj...
 5.6.21: To make sure that a room is rectangular, builders check the two dia...
 5.6.22: The platforms shown at the beginning of this lesson lift objects st...
 5.6.23: At the street intersection shown at right, one of the streets is wi...
 5.6.24: In Exercises 24 and 25, use only the two parallel edges of your dou...
 5.6.25: In Exercises 24 and 25, use only the two parallel edges of your dou...
 5.6.26: Developing Proof Complete the flowchart proof below to demonstrate ...
 5.6.27: Developing Proof In the last exercise, you proved that if the four ...
 5.6.28: Trace the figure below. Calculate the measure of each lettered angle.
 5.6.29: Find the coordinates of three more points that lie on the line pass...
 5.6.30: Write the equation of the perpendicular bisector of the segment wit...
 5.6.31: ABC has vertices A(0, 0), B(4, 2), and C(8, 8). What is the equatio...
 5.6.32: Construction Oran Boatwright is rowing at a 60 angle from the upstr...
Solutions for Chapter 5.6: Properties of Special Parallelograms
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 5.6: Properties of Special Parallelograms
Get Full SolutionsSince 32 problems in chapter 5.6: Properties of Special Parallelograms have been answered, more than 23286 students have viewed full stepbystep solutions from this chapter. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Chapter 5.6: Properties of Special Parallelograms includes 32 full stepbystep solutions. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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 •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Solvable system Ax = b.
The right side b is in the column space of A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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