 8.1: \im/LR = 30, then mARSU = _Z_,m L. TSU = _Z_, and m LT =
 8.2: If m R = k, then m L RSU = _Z_,m LTSU = _2_, and m LT = :_.
 8.3: ARST ~ A_l_ ~ A^
 8.4: A/?5f/ ~ A. A.
 8.5: Simplify. 50
 8.6: Simplify.3V8
 8.7: Simplify.V225
 8.8: Simplify.7V63
 8.9: Simplify.288
 8.10: Simplify10
 8.11: Simplify./I
 8.12: Simplify.5
 8.13: Simplify.A
 8.14: Simplify.^ /??
 8.15: Give the geometric mean between:a. 2 and 3 b. 2 and 6 c. 4 and 25
 8.16: a. / is the geometric mean between ? and _b. u is the geometric mea...
 8.17: a. z is the geometric mean between ? andThus z = _L_.b. x is the ge...
 8.18: Find the geometric mean between the two numbers. 49 and 25
 8.19: Find the geometric mean between the two numbers. 1 and 1000
 8.20: Find the geometric mean between the two numbers. 16 and 24
 8.21: Find the geometric mean between the two numbers. 22 and 55
 8.22: If LA/ = 4 and MK = 8, find JM.
 8.23: If LM = 6 and JM = 4, find MK.
 8.24: If JM = 3 and MK = 6, find LM.
 8.25: If JM = 4 and JA: = 9, find L/C.
 8.26: If JM = 3 and MK = 9, find L7.
 8.27: If JM = 3 and JL = 6, find MK.
 8.28: If JL = 9 and JM = 6, find MK.
 8.29: If L/C = 3V6 and MK = 6, find JM.
 8.30: If LA" = 7 and MA" = 6, find JM.
 8.31: Find the values of x, v, and z.
 8.32: Find the values of x, v, and z.
 8.33: Find the values of x, v, and z.
 8.34: Find the values of x, v, and z.
 8.35: Find the values of x, v, and z.
 8.36: Find the values of x, v, and z.
 8.37: Find the values of x, v, and z.
 8.38: Find the values of x, v, and z.
 8.39: Find the values of x, v, and z.
 8.40: Prove Theorem 81.
 8.41: a. Refer to the figure at the right, and use Corollary 2to complete...
 8.42: Prove: In a right triangle, the product of the hypotenuse and the l...
 8.43: Given: PQRS is a rectangle;PS is the geometric meanbetween 57" and ...
 8.44: Given: PQRS is a rectangle;ZA is a right angle.Prove: BS RC = PS QR...
 8.45: The arithmetic mean between two numbers r and s is defined to bea. ...
Solutions for Chapter 8: Similarity in Right Triangles
Full solutions for Geometry  1st Edition
ISBN: 9780395977279
Solutions for Chapter 8: Similarity in Right Triangles
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 45 problems in chapter 8: Similarity in Right Triangles have been answered, more than 5388 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Geometry, edition: 1. Geometry was written by and is associated to the ISBN: 9780395977279. Chapter 8: Similarity in Right Triangles includes 45 full stepbystep solutions.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(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.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.