 51.1: Vocabulary A ? is the locus of all points in a plane that are equid...
 51.2: Use the diagram for Exercises 24.Given that PS = 53.4, QT = 47.7, a...
 51.3: Use the diagram for Exercises 24.Given that m is the perpendicular ...
 51.4: Use the diagram for Exercises 24.Given that m is the perpendicular ...
 51.5: Use the diagram for Exercises 57.Given that BD bisects ABC and CD =...
 51.6: Use the diagram for Exercises 57.Given that AD = 61, CD = 61, and m...
 51.7: Use the diagram for Exercises 57.Given that DA = DC, mDBC = (10y + ...
 51.8: Carpentry For a king post truss to be constructed correctly, P must...
 51.9: Write an equation in pointslope form forthe perpendicular bisector...
 51.10: Write an equation in pointslope form forthe perpendicular bisector...
 51.11: Write an equation in pointslope form forthe perpendicular bisector...
 51.12: Use the diagram for Exercises 1214.Given that line t is the perpend...
 51.13: Use the diagram for Exercises 1214.Given that line t is the perpend...
 51.14: Use the diagram for Exercises 1214.Given that GJ = 70.2, JH = 26.5,...
 51.15: Use the diagram for Exercises 1517.Given that mRSQ = mTSQ and TQ = ...
 51.16: Use the diagram for Exercises 1517.Given that mRSQ = 58, RQ = 49, a...
 51.17: Use the diagram for Exercises 1517.Given that RQ = TQ, mQSR = (9a +...
 51.18: City Planning The planners for a new section of the city want every...
 51.19: Write an equation in pointslope form for the perpendicularbisector...
 51.20: Write an equation in pointslope form for the perpendicularbisector...
 51.21: Write an equation in pointslope form for the perpendicularbisector...
 51.22: PQ is the perpendicular bisector of ST . Find the values of m and n.
 51.23: Shuffleboard Use the diagram of a shuffleboard and the following in...
 51.24: Shuffleboard Use the diagram of a shuffleboard and the following in...
 51.25: Shuffleboard Use the diagram of a shuffleboard and the following in...
 51.26: Shuffleboard Use the diagram of a shuffleboard and the following in...
 51.27: Shuffleboard Use the diagram of a shuffleboard and the following in...
 51.28: Shuffleboard Use the diagram of a shuffleboard and the following in...
 51.29: MultiStep The endpoints of AB are A (2, 1)and B (4, 3) . Find th...
 51.30: Write a paragraph proof of the Converse of thePerpendicular Bisecto...
 51.31: Write a twocolumn proof of the Angle Bisector Theorem. Given: PS b...
 51.32: Critical Thinking In the Converse of the Angle Bisector Theorem, wh...
 51.33: This problem will prepare you for the Concept Connection on page 32...
 51.34: Write About It How is the construction of the perpendicular bisecto...
 51.35: If JK is perpendicular to XY at its midpoint M, which statement is ...
 51.36: What information is needed to conclude that EF is the bisector of D...
 51.37: Short Response The city wants to build a visitor centerin the park ...
 51.38: Consider the points P (2, 0) , A (4, 2) , B (0, 6) , and C (6, 3...
 51.39: Find the locus of points that are equidistant from the xaxis and y...
 51.40: Write a twocolumn proof of the Converse of the Angle Bisector Theo...
 51.41: Write a paragraph proof.Given:KN is the perpendicular bisector of J...
 51.42: Lyn bought a sweater for $16.95. The change c that she received can...
 51.43: For the points R (4, 2) , S (1, 4) , T (3, 1) , and V (7, 5) , ...
 51.44: For the points R (4, 2) , S (1, 4) , T (3, 1) , and V (7, 5) , ...
 51.45: For the points R (4, 2) , S (1, 4) , T (3, 1) , and V (7, 5) , ...
 51.46: Write the equation of each line in slopeintercept form. (Lesson 3...
 51.47: Write the equation of each line in slopeintercept form. (Lesson 3...
 51.48: Write the equation of each line in slopeintercept form. (Lesson 3...
Solutions for Chapter 51: Perpendicular and Angle Bisectors
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 51: Perpendicular and Angle Bisectors
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 51: Perpendicular and Angle Bisectors includes 48 full stepbystep solutions. Geometry was written by and is associated to the ISBN: 9780030923456. This textbook survival guide was created for the textbook: Geometry, edition: 1. Since 48 problems in chapter 51: Perpendicular and Angle Bisectors have been answered, more than 41838 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

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 ().

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

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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