 61.1: Vocabulary Explain why an equilateral polygon is not necessarily a ...
 61.2: Tell whether each outlined shape is a polygon. If it is a polygon, ...
 61.3: Tell whether each outlined shape is a polygon. If it is a polygon, ...
 61.4: Tell whether each outlined shape is a polygon. If it is a polygon, ...
 61.5: Tell whether each outlined shape is a polygon. If it is a polygon, ...
 61.6: Tell whether each polygon is regular or irregular. Tell whether it ...
 61.7: Tell whether each polygon is regular or irregular. Tell whether it ...
 61.8: Tell whether each polygon is regular or irregular. Tell whether it ...
 61.9: Find the measure of each interior angle of pentagon ABCDE.
 61.10: Find the measure of each interior angle of a regular dodecagon.
 61.11: Find the sum of the interior anglemeasures of a convex 20gon
 61.12: Find the value of y in polygon JKLM
 61.13: Find the measure of each exterior angle of a regular pentagon.
 61.14: Safety Use the photograph of the traffic sign forExercises 14 and 1...
 61.15: Safety Use the photograph of the traffic sign forExercises 14 and 1...
 61.16: Tell whether each figure is a polygon. If it is a polygon, name it ...
 61.17: Tell whether each figure is a polygon. If it is a polygon, name it ...
 61.18: Tell whether each figure is a polygon. If it is a polygon, name it ...
 61.19: Tell whether each polygon is regular or irregular. Tell whether it ...
 61.20: Tell whether each polygon is regular or irregular. Tell whether it ...
 61.21: Tell whether each polygon is regular or irregular. Tell whether it ...
 61.22: Find the measure of each interior angle of quadrilateral RSTV.
 61.23: Find the measure of each interior angle of a regular 18gon.
 61.24: Find the sum of the interior angle measures of a convex heptagon.
 61.25: Find the measure of each exterior angle of a regular nonagon.
 61.26: A pentagon has exterior angle measures of5a, 4a, 10a, 3a, and 8a. F...
 61.27: Crafts The folds on the lid of the gift box form a regular hexagon....
 61.28: Crafts The folds on the lid of the gift box form a regular hexagon....
 61.29: Algebra Find the value of x in each figure.29
 61.30: Algebra Find the value of x in each figure.30
 61.31: Algebra Find the value of x in each figure.31
 61.32: Find the number of sides a regular polygon must have to meet each c...
 61.33: Find the number of sides a regular polygon must have to meet each c...
 61.34: Find the number of sides a regular polygon must have to meet each c...
 61.35: Name the convex polygon whose interior angle measures have each giv...
 61.36: Name the convex polygon whose interior angle measures have each giv...
 61.37: Name the convex polygon whose interior angle measures have each giv...
 61.38: Name the convex polygon whose interior angle measures have each giv...
 61.39: MultiStep An exterior angle measure of a regular polygon is given....
 61.40: MultiStep An exterior angle measure of a regular polygon is given....
 61.41: MultiStep An exterior angle measure of a regular polygon is given....
 61.42: MultiStep An exterior angle measure of a regular polygon is given....
 61.43: /////ERROR ANALYSIS///// Which conclusion is incorrect?Explain the ...
 61.44: Estimation Graph the polygon formed by the points A (2, 6) , B (...
 61.45: This problem will prepare you for the Concept Connection on page 40...
 61.46: The perimeter of a regular polygon is 45 inches. The length of one ...
 61.47: Draw an example of each figure.a regular quadrilateral
 61.48: Draw an example of each figure.an irregular concave heptagon
 61.49: Draw an example of each figure.an irregular convex pentagon
 61.50: Draw an example of each figure.an equilateral polygon that is not e...
 61.51: Write About It Use the terms from the lesson to describethe figure ...
 61.52: Critical Thinking What geometric figure does a regularpolygon begin...
 61.53: Which terms describe the figure shown?I. quadrilateral II. concave ...
 61.54: Which statement is NOT true about a regular 16gon? It is a convex ...
 61.55: In polygon ABCD, mA = 49, mB = 107, and mC = 2mD. What is mC? 24 68...
 61.56: The interior angle measures of a convex pentagon are consecutive mu...
 61.57: Polygon PQRST is a regular pentagon. Find the values of x, y, and z
 61.58: MultiStep Polygon ABCDEFGHJK is a regular decagon. Sides AB and DE...
 61.59: Critical Thinking Does the Polygon Angle Sum Theorem workfor concav...
 61.60: Solve by factoring. (Previous course)x 2 + 3x  10 = 0
 61.61: Solve by factoring. (Previous course)x 2  x  12 = 0
 61.62: Solve by factoring. (Previous course)x 2  12x = 35
 61.63: The lengths of two sides of a triangle are given. Find the range of...
 61.64: The lengths of two sides of a triangle are given. Find the range of...
 61.65: The lengths of two sides of a triangle are given. Find the range of...
 61.66: Find each side length for a 306090 triangle. (Lesson 58)the leng...
 61.67: Find each side length for a 306090 triangle. (Lesson 58)the leng...
Solutions for Chapter 61: Properties and Attributes of Polygons
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 61: Properties and Attributes of Polygons
Get Full SolutionsGeometry was written by and is associated to the ISBN: 9780030923456. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Geometry, edition: 1. Chapter 61: Properties and Attributes of Polygons includes 67 full stepbystep solutions. Since 67 problems in chapter 61: Properties and Attributes of Polygons have been answered, more than 43710 students have viewed full stepbystep solutions from this chapter.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues 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].

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

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.

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 ().

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as 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.

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.

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

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