 101.1: (5, 6), (1, 7)
 101.2: (8, 9), (3, 4.5)
 101.3: (13, 4), (10, 14.6)
 101.4: (12, 2), (3.5, 7)
 101.5: (2, 4), (10, 10)
 101.6: (7, 8), (4, 9)
 101.7: (0.5, 1.4), (1.1, 2.9)
 101.8: (4.3, 2.6), (6.5, 3.4)
 101.9: STANDARDIZED TEST PRACTICE The map of a mall is overlaid with a num...
 101.10: (8, 3), (16, 7)
 101.11: (5, 3), (3, 7)
 101.12: (6, 5), (2, 7)
 101.13: (5, 9), (12, 18)
 101.14: GEOMETRY Triangle MNP has vertices M(3, 5), N(2, 8), and P(7, 4)....
 101.15: REAL ESTATE In Johns town, the numbered streets and avenues form a ...
 101.16: (4, 9), (1, 3)
 101.17: (1, 14), (6, 10)
 101.18: (4, 10), (3, 11)
 101.19: (9, 2), (12, 14)
 101.20: (0.23, 0.4), (0.68, 0.2)
 101.21: (2.3, 1.2), (4.5, 3.7)
 101.22: GEOMETRY Quadrilateral RSTV has vertices R(4, 6), S(4, 5), T(6, 3)...
 101.23: GEOMETRY Triangle BCD has vertices B(4, 9), C(8, 9), and D(6, 5)....
 101.24: (3, _2 11), (5, _9 11)
 101.25: (0, _1 5), (_3 5 , _3 5)
 101.26: (2 3 , 5), (3 3 , 9)
 101.27: (_23 3 , _ 5 4 ), (_2 3 3 , _ 5 2 )
 101.28: GEOMETRY Find the perimeter and area of the triangle at the right.
 101.29: GEOMETRY A circle has a radius with endpoints at (2, 5) and (1, 4...
 101.30: GEOMETRY Circle Q has a diameter AB. If A is at (3, 5) and the ce...
 101.31: Approximate the center of the United States. Describe your method.
 101.32: RESEARCH Use the Internet or other reference to look up the USGS ge...
 101.33: How far is it from Birmingham to Montgomery if each unit on the gri...
 101.34: How long would it take a plane to fly from Huntsville to Montgomery...
 101.35: WOODWORKING A stage crew is making the set for a childrens play. Th...
 101.36: OPEN ENDED Find two points that are 29 units apart.
 101.37: REASONING Identify all of the points that are equidistant from the ...
 101.38: CHALLENGE Verify the Midpoint Formula. (Hint: You must show that th...
 101.39: Writing in Math Explain how to use the Distance Formula to approxim...
 101.40: ACT/SAT Point D(5, 1) is the midpoint of segment CE. If point C ha...
 101.41: REVIEW If log10 x = 3, what is the value of x? F x = 1000 H x = _1...
 101.42: COMPUTERS Suppose a computer that costs $3000 new is only worth $60...
 101.43: 3ex  2 = 0
 101.44: e3x = 4
 101.45: ln (x + 2) = 5
 101.46: y = x2 + 6x + 9
 101.47: y = 2x2 + 20x + 50
 101.48: y = 3x2  18x  10
Solutions for Chapter 101: Midpoint and Distance Formulas
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 101: Midpoint and Distance Formulas
Get Full SolutionsAlgebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Chapter 101: Midpoint and Distance Formulas includes 48 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 48 problems in chapter 101: Midpoint and Distance Formulas have been answered, more than 53950 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Covariance matrix:E.
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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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!).

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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.