 8.6.1: In Exercises 18, find the area of the shaded region. The radius of ...
 8.6.2: In Exercises 18, find the area of the shaded region. The radius of ...
 8.6.3: In Exercises 18, find the area of the shaded region. The radius of ...
 8.6.4: In Exercises 18, find the area of the shaded region. The radius of ...
 8.6.5: In Exercises 18, find the area of the shaded region. The radius of ...
 8.6.6: In Exercises 18, find the area of the shaded region. The radius of ...
 8.6.7: In Exercises 18, find the area of the shaded region. The radius of ...
 8.6.8: In Exercises 18, find the area of the shaded region. The radius of ...
 8.6.9: The shaded area is 12 cm2. Find r
 8.6.10: The shaded area is 32 cm2. Find r.
 8.6.11: The shaded area is 120 cm2, and the radius is 24 cm. Find x.
 8.6.12: The shaded area is 10 cm2 The radius of the large circle is 10 cm, ...
 8.6.13: Application Suppose the pizza slice in the photo at the beginning o...
 8.6.14: Utopia Park has just installed a circular fountain 8 meters in diam...
 8.6.15: The illustrations below demonstrate how to find a rectangle with th...
 8.6.16: Construction Reverse the process you used in Exercise 15. On graph ...
 8.6.17: Each set of circles is externally tangent. What is the area of the ...
 8.6.18: The height of a trapezoid is 15 m and the midsegment is 32 m. What ...
 8.6.19: CE, BH,and AG are altitudes. Find AB and AG.
 8.6.20: In Exercises 2023, identify each statement as true or false. If tru...
 8.6.21: In Exercises 2023, identify each statement as true or false. If tru...
 8.6.22: In Exercises 2023, identify each statement as true or false. If tru...
 8.6.23: In Exercises 2023, identify each statement as true or false. If tru...
Solutions for Chapter 8.6: Any Way You Slice It
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 8.6: Any Way You Slice It
Get Full SolutionsSince 23 problems in chapter 8.6: Any Way You Slice It have been answered, more than 22111 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Chapter 8.6: Any Way You Slice It includes 23 full stepbystep solutions. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4.

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

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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

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

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

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].

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.