 64.1: Vocabulary What is another name for an equilateral quadrilateral? a...
 64.2: Engineering The braces of the bridge supportlie along the diagonals...
 64.3: Engineering The braces of the bridge supportlie along the diagonals...
 64.4: Engineering The braces of the bridge supportlie along the diagonals...
 64.5: Engineering The braces of the bridge supportlie along the diagonals...
 64.6: ABCD is a rhombus. Find each measure. AB
 64.7: ABCD is a rhombus. Find each measure. mABC
 64.8: MultiStep The vertices of square JKLMare J (3, 5) , K (4, 1) , ...
 64.9: Given: RECT is a rectangle. RX TY Prove: REY TCX
 64.10: Carpentry A carpenter measures the diagonals ofa piece of wood. In ...
 64.11: Carpentry A carpenter measures the diagonals ofa piece of wood. In ...
 64.12: Carpentry A carpenter measures the diagonals ofa piece of wood. In ...
 64.13: Carpentry A carpenter measures the diagonals ofa piece of wood. In ...
 64.14: VWXY is a rhombus. Find each measure.VW
 64.15: VWXY is a rhombus. Find each measure.mVWX and mWYX ifmWVY = (4b + 1...
 64.16: MultiStep The vertices of square PQRS are P (4, 0) , Q (4, 3) , R...
 64.17: Given: RHMB is a rhombus with diagonal HB . Prove: HMX HRX
 64.18: Find the measures of the numbered angles in each rectangle.18
 64.19: Find the measures of the numbered angles in each rectangle.19
 64.20: Find the measures of the numbered angles in each rectangle.20
 64.21: Find the measures of the numbered angles in each rhombus.21
 64.22: Find the measures of the numbered angles in each rhombus.22
 64.23: Find the measures of the numbered angles in each rhombus.23
 64.24: Tell whether each statement is sometimes, always, or never true.(Hi...
 64.25: Tell whether each statement is sometimes, always, or never true.(Hi...
 64.26: Tell whether each statement is sometimes, always, or never true.(Hi...
 64.27: Tell whether each statement is sometimes, always, or never true.(Hi...
 64.28: Tell whether each statement is sometimes, always, or never true.(Hi...
 64.29: Tell whether each statement is sometimes, always, or never true.(Hi...
 64.30: Tell whether each statement is sometimes, always, or never true.(Hi...
 64.31: Tell whether each statement is sometimes, always, or never true.(Hi...
 64.32: Critical Thinking A triangle is equilateral if and only if the tria...
 64.33: History There are five shapes of clay tiles inthis tile mosaic from...
 64.34: /////ERROR ANALYSIS///// Find and correctthe error in this proof of...
 64.35: Complete the twocolumn proof of Theorem 642 by filling in the bl...
 64.36: This problem will prepare you for the Concept Connection on page 43...
 64.37: Use this plan to write a proof of Theorem 644. Given: VWXY is a r...
 64.38: Write a paragraph proof of Theorem 641. Given: ABCD is a rectangl...
 64.39: Write a twocolumn proof. Given: ABCD is a rhombus. E, F, G, andH a...
 64.40: MultiStep Find the perimeter and area of each figure. Round to the...
 64.41: MultiStep Find the perimeter and area of each figure. Round to the...
 64.42: MultiStep Find the perimeter and area of each figure. Round to the...
 64.43: Write About It Explain why each of these conditional statements is ...
 64.44: Write About It List the properties that a square inherits because i...
 64.45: Which expression represents the measure of J in rhombus JKLM?x (180...
 64.46: Short Response The diagonals of rectangle QRST intersect at point P...
 64.47: Which statement is NOT true of a rectangle? Both pairs of opposite ...
 64.48: Algebra Find the value of x in the rhombus.
 64.49: Prove that the segment joining the midpoints oftwo consecutive side...
 64.50: Extend the definition of a triangle midsegment to write a definitio...
 64.51: The figure is formed by joining eleven congruent squares.How many r...
 64.52: The cost c of a taxi ride is given by c = 2 + 1.8 (m  1) , where m...
 64.53: Determine if each conditional is true. If false, give a counterexam...
 64.54: Determine if each conditional is true. If false, give a counterexam...
 64.55: Determine if each quadrilateral must be a parallelogram. Justify yo...
 64.56: Determine if each quadrilateral must be a parallelogram. Justify yo...
Solutions for Chapter 64: Properties of Special Parallelograms
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 64: Properties of Special Parallelograms
Get Full SolutionsGeometry was written by and is associated to the ISBN: 9780030923456. This textbook survival guide was created for the textbook: Geometry, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 56 problems in chapter 64: Properties of Special Parallelograms have been answered, more than 42177 students have viewed full stepbystep solutions from this chapter. Chapter 64: Properties of Special Parallelograms includes 56 full stepbystep solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column space C (A) =
space of all combinations of the columns of 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].

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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 •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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