 10.1: Given: AJKMExplain how to construct a triangle that iscongruent to ...
 10.2: Draw any AB.a. Construct XY so that XY = AB.b. Using X and Y as cen...
 10.3: Explain how you could construct a 30 angle.
 10.4: Exercise 3 suggests that you could construct other angles with cert...
 10.5: Suppose you are given the three lengths shown and are rasked to con...
 10.6: Z 1 and Z2 are given. You see two attempts at constructing an angle...
 10.7: A square whose sides each have length AC
 10.8: A square whose perimeter equals AC
 10.9: A right triangle with hypotenuse and one leg equal to AC and BC,res...
 10.10: A triangle whose sides are in the ratio 1 :2:\/3
 10.11: Where does the center of the circle lie with respect toline / and p...
 10.12: Where does the center of the circle lie with respect toAY?
 10.13: Explain how to carry out the construction of the circle.
 10.14: a. Draw an acute triangle. Construct the perpendicular bisector of ...
 10.15: a. Draw an acute triangle. Construct the three altitudes.b. Do the ...
 10.16: a. Draw a very large acute triangle. Construct the three medians.b....
 10.17: A parallelogram with an n angle and sides of lengths a and b
 10.18: A rectangle with sides of lengths a and b
 10.19: A square with perimeter 2a
 10.20: A rhombus with diagonals of lengths a and b
 10.21: A square with diagonals of length
 10.22: A segment of length \Ja2 + b2
 10.23: A square with diagonals of length b\J2
 10.24: A right triangle with hypotenuse of length a and one leg of length b
 10.25: Drawji segment and let its length be s. Construct a segment whose l...
 10.26: Draw a diagram roughly like the one shown. Without laying your stra...
 10.27: Draw three noncollinear points R, S, and T. Construct a triangle wh...
 10.28: Draw a segment and let its length be 1.a. Construct a segment of le...
Solutions for Chapter 10: What Construction Means
Full solutions for Geometry  1st Edition
ISBN: 9780395977279
Solutions for Chapter 10: What Construction Means
Get Full SolutionsChapter 10: What Construction Means includes 28 full stepbystep solutions. Geometry was written by and is associated to the ISBN: 9780395977279. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Geometry, edition: 1. Since 28 problems in chapter 10: What Construction Means have been answered, more than 5395 students have viewed full stepbystep solutions from this chapter.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.