- 8.4.1: In Exercises 18, use the Regular Polygon Area Conjecture to find th...
- 8.4.2: In Exercises 18, use the Regular Polygon Area Conjecture to find th...
- 8.4.3: In Exercises 18, use the Regular Polygon Area Conjecture to find th...
- 8.4.4: In Exercises 18, use the Regular Polygon Area Conjecture to find th...
- 8.4.5: In Exercises 18, use the Regular Polygon Area Conjecture to find th...
- 8.4.6: In Exercises 18, use the Regular Polygon Area Conjecture to find th...
- 8.4.7: In Exercises 18, use the Regular Polygon Area Conjecture to find th...
- 8.4.8: In Exercises 18, use the Regular Polygon Area Conjecture to find th...
- 8.4.9: Construction Draw a segment 4 cm long. Use a compass and straighted...
- 8.4.10: Draw a regular pentagon with apothem 4 cm. Use the Regular Polygon ...
- 8.4.11: A square is also a regular polygon. How is the apothem of a square ...
- 8.4.12: Technology Use geometry software to construct a circle. Inscribe a ...
- 8.4.13: Find the approximate area of the shaded region of the regular octag...
- 8.4.14: Find the approximate area of the shaded regular hexagonal donut. Th...
- 8.4.15: Application An interior designer created the kitchen plan shown. Th...
- 8.4.16: In Exercises 16 and 17, graph the two lines, then find the area bou...
- 8.4.17: In Exercises 16 and 17, graph the two lines, then find the area bou...
- 8.4.18: Technology Construct a triangle and its three medians. Compare the ...
- 8.4.19: If the pattern continues, write an expression for the perimeter of ...
- 8.4.20: GHJK is a rectangle. Find the area of pentagon GHIJK.
- 8.4.21: FELA and CDLB are parallelograms. Find the area of the shaded region.
Solutions for Chapter 8.4: Areas of Regular Polygons
Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
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!
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
cond(A) = c(A) = IIAIlIIA-III = 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.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
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.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).