- 5.1: How do you find the measure of one exterior angle of a regular poly...
- 5.2: How can you find the number of sides of an equiangular polygon by m...
- 5.3: How do you construct a rhombus by using only a ruler or double-edge...
- 5.4: How do you bisect an angle by using only a ruler or double-edged st...
- 5.5: How can you use the converse of the Rectangle Diagonals Conjecture ...
- 5.6: How can you use the Triangle Midsegment Conjecture to find a distan...
- 5.7: Find x and y.
- 5.8: The perimeter is 266 cm. Find x.
- 5.9: Find a and c.
- 5.10: MS is a midsegment. Find the perimeter of MOIS.
- 5.11: Find x.
- 5.12: Find y and z.
- 5.13: Copy and complete the table below by placing a yes (to mean always)...
- 5.14: Application A 2-inch-wide frame is to be built around the regular d...
- 5.15: Developing Proof Find the measure of each lettered angle. Explain h...
- 5.16: Archaeologist Ertha Diggs has uncovered one stone that appears to b...
- 5.17: Kite ABCD has vertices A(3, 2), B(2, 2), C(3, 1), and D(0, 2). Find...
- 5.18: When you swing left to right on a swing, the seat stays parallel to...
- 5.19: Construction The tiling of congruent pentagons shown below is creat...
- 5.20: Construction An airplane is heading north at 900 km/h. However, a 5...
- 5.21: Construction In Exercises 2124, use the given segments and angles t...
- 5.22: Construction In Exercises 2124, use the given segments and angles t...
- 5.23: Construction In Exercises 2124, use the given segments and angles t...
- 5.24: Construction In Exercises 2124, use the given segments and angles t...
- 5.25: Three regular polygons meet at point B. Only four sides of the thir...
- 5.26: Find x
- 5.27: Developing Proof Prove the conjecture: The diagonals of a rhombus b...
- 5.28: Developing Proof Use a labeled diagram to prove the Parallelogram O...
Solutions for Chapter 5: Discovering and Proving Polygon Properties
Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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!).
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
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.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
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.
Every v in V is orthogonal to every w in W.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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·
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.