- 11-4.1: Vocabulary A, B, and C lie on P. ABC is an example of an ? angle.(i...
- 11-4.2: Find each measure.mDEF
- 11-4.3: Find each measure.m EG
- 11-4.4: Find each measure.m JKL
- 11-4.5: Find each measure.mLKM
- 11-4.6: Crafts A circular loom can be used for knitting. What is the mQTR i...
- 11-4.7: Find each valuex
- 11-4.8: Find each valuey
- 11-4.9: Find each valuemXYZ
- 11-4.10: Multi-Step Find the angle measures of each quadrilateral.PQRS
- 11-4.11: Multi-Step Find the angle measures of each quadrilateral.ABCD
- 11-4.12: Find each measure.m ML
- 11-4.13: Find each measure.mKMN
- 11-4.14: Find each measure.m EGH
- 11-4.15: Find each measure.mGFH
- 11-4.16: rafts An artist created a stainedglass window. If mBEC = 40 andm AB...
- 11-4.30: Given: ABC is inscribed in X with X in the interior of ABC.Prove: m...
- 11-4.31: Given: ABC is inscribed in X with X in the exterior of ABC.Prove: m...
- 11-4.32: Prove Corollary 11-4-2. Given: ACB and ADB intercept AB .Prove: ACB...
- 11-4.33: Multi-Step In the diagram, m JKL = 198, and m KLM = 216. Find the m...
- 11-4.34: Critical Thinking A rectangle PQRS is inscribedin a circle. What ca...
- 11-4.35: History The diagram shows the Winchester RoundTable with inscribed ...
- 11-4.36: To inscribe an equilateral triangle in a circle, draw a diameter BC...
- 11-4.37: Write About It A student claimed that if a parallelogramcontains a ...
- 11-4.38: Construction Circumscribe a circle about a triangle. (Hint: Follow ...
- 11-4.39: What is mBAC? 38 66 43 81
- 11-4.40: Equilateral XCZ is inscribed in a circle. If CY bisects C, what is ...
- 11-4.41: Quadrilateral ABCD is inscribed in a circle. The ratio ofmA to mC i...
- 11-4.42: Which of these angles has the greatest measure? STR QPR QSR PQS
- 11-4.43: Prove that an inscribed angle subtends a semicircle if and only if ...
- 11-4.44: Prove that if a quadrilateral is inscribed in a circle, then its op...
- 11-4.45: Find m PQ to thenearest degree.
- 11-4.46: Find mABD
- 11-4.47: Construction To circumscribe an equilateral triangle about a circle...
- 11-4.48: Tickets for a play cost $15.00 for section C, $22.50 for section B,...
- 11-4.49: Write a ratio expressing the slope of the line through each pair of...
- 11-4.50: Write a ratio expressing the slope of the line through each pair of...
- 11-4.51: Write a ratio expressing the slope of the line through each pair of...
- 11-4.52: Find each of the following. (Lesson 11-2)m ST
- 11-4.53: Find each of the following. (Lesson 11-2)area of ABD
Solutions for Chapter 11-4: Inscribed Angles
Full solutions for Geometry | 1st Edition
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!
peA) = det(A - AI) has peA) = zero matrix.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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.
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.