 1.6.1: Based on the marks, what can you assume to be true in each figure?
 1.6.2: For Exercises 26, match the term on the left with its figure on the...
 1.6.3: For Exercises 26, match the term on the left with its figure on the...
 1.6.4: For Exercises 26, match the term on the left with its figure on the...
 1.6.5: For Exercises 26, match the term on the left with its figure on the...
 1.6.6: For Exercises 26, match the term on the left with its figure on the...
 1.6.7: For Exercises 710, sketch and label the figure. Mark the figures. T...
 1.6.8: For Exercises 710, sketch and label the figure. Mark the figures. K...
 1.6.9: For Exercises 710, sketch and label the figure. Mark the figures. R...
 1.6.10: For Exercises 710, sketch and label the figure. Mark the figures. R...
 1.6.11: Draw a hexagon with exactly two outside diagonals.
 1.6.12: Draw a regular quadrilateral. What is another name for this shape?
 1.6.13: Find the other two vertices of a square with one vertex (0, 0) and ...
 1.6.14: A rectangle with perimeter 198 cm is divided into five congruent re...
 1.6.15: For Exercises 1518, copy the given polygon and segment onto graph p...
 1.6.16: For Exercises 1518, copy the given polygon and segment onto graph p...
 1.6.17: For Exercises 1518, copy the given polygon and segment onto graph p...
 1.6.18: For Exercises 1518, copy the given polygon and segment onto graph p...
 1.6.19: Draw and cut out two congruent acute scalene triangles. a. Arrange ...
 1.6.20: Draw and cut out two congruent obtuse isosceles triangles. Which sp...
 1.6.21: Imagine using two congruent triangles to create a special quadrilat...
 1.6.22: For Exercises 2224, sketch and carefully label the figure. Mark the...
 1.6.23: For Exercises 2224, sketch and carefully label the figure. Mark the...
 1.6.24: For Exercises 2224, sketch and carefully label the figure. Mark the...
 1.6.25: Draw an equilateral octagon ABCDEFGH with A(5, 0), B(4, 4), and C(0...
Solutions for Chapter 1.6: Special Quadrilaterals
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 1.6: Special Quadrilaterals
Get Full SolutionsChapter 1.6: Special Quadrilaterals includes 25 full stepbystep solutions. Since 25 problems in chapter 1.6: Special Quadrilaterals have been answered, more than 23393 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their 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.

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Fundamental Theorem.
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.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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