Chapter 1.2, Problem GEE
Step 1 0f 15 A
We wish to show that the perpendicular bisectprs of the three sides of a triangle are
concurrent,
Com ment
Step 2 of 15 A
[t A(z.y.). Man). and C(lel be the vertlces 01 the triangle, Then the
perpendicular bisectors
EE Chapter 1.2, Pmbbem GEE D Bookmark Showallsteps: m a:
= (xwz am my. '13.": Hay. in)
If. y. l
4: y: l
x, y. l
=44.
Com ment
Step 9 of 15 A
Note that detd \s nonzero since the vemces A, B, and C are not collinear. Thus we see that
A 5 mvertibeand
144:;
Chapter 1.2, Problem GEE
D Bookmark Showallsteps: LIE- :
perpendicular bisectors Ll of EC. [1 cl CA . and L] of AB are given bythe fallawmg
equations:
11=cfw_[w)ER=(r-r=+(y-yzl1=(1-13)2+(yya]::
L, =[iayiew=(x-x.)+iyy.i =ix-x.)+(y-y.i=
L; =l(w) 11 11-h): +
Chaptel 1.2, Problem GEE
19-232 +xzz+y-2m+yz-x-2u,+xa'+y-2m+y. -
Step 4 of 15 A
Subtracting the right side from the left yields
2m, -X:)+2y(y. -y:)+1: -x3 +ya m =0-
whence
24: _x1)+2)'(yl y1)=x): 33:] +95: _y21
Step 5 of 15 A
Similarly, Simplifying the eq
55 Chapter 1.2, Problem GEE D Bookmark Showallsteps: m :
Slepll Df 15 A
Similarlywe have
y=i[(.'l|2 71:(1, 7x)*(yxl 711. II)+(J"JI 7x12)"l 7%)]-
Slep 13 of15 A
Now ifmcnmpute the coordinata x' and y' of lntersection point (y') 01 L, and LI,
we obtain the
55 Chapter 1.2, Problem GEE I] Bonkmark Show all steps: a:
Problem
A perpendicular bisecmr at a line segment is a line through the midpoint of the segment,
perpendicular to the segment (Figure 149), Prnve that the perpendicular hisectors of the
three side
Chaptel 1.2, Problem GEE
Mar. -X.)+2y(y. -y3)=X. -x, +y. -y]A
Com ment
Step 6 of 15 A
Thus, the intersectinnoi L. and L, is the solution (1?) to the lollcmhg linear system:
2:"(xJ x,)+2y(y, y,)= x, x,: +y, y,;
21.13:. -x.)+2y(.v. -.v:)=x. -xs +y.' m
Ste
55 Chapter 1.2, Problem GEE l1 Buokmark Showasteps: LEE- :5
Slepl of 15 A
Now ifmcnmpute the coordinates x' and y' of intersection point (1'0) 01 L, and 1.
we obtain the same formulas m fur x andy with the indices cyclically permuted, Thus we
have
[(12' W
MATH 110 QUIZ 5
AUGUST 7, 2014
NAME:
No notes or books are allowed on the following quiz. Please justify all of your answers
unless indicated otherwise.
1. Consider C3 with the standard inner product. Determine whether each of the following linear operato
MATH 110 HWK 7: DUE WEDNESDAY, AUGUST 6, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any
of the results in Chapters 1, 2, 3, 4, 5, 6, 7.A-7.B, any of the handouts on the website, or
previously solved homework pro
MATH 110 HWK 3 SOLUTIONS
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2 and 3.A-D, the Fields and Polynomials supplementary reading handouts, or previously solved
homework problems wit
MATH 110 HWK 5: DUE WEDNESDAY, JULY 23, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2, 3, 4, 5, any of the handouts on the website, or previously solved homework problems without
MATH 110 HWK 4: DUE WEDNESDAY, JULY 16, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2, 3, 4, 5.A, any of the handouts on the website, or previously solved homework problems witho
MATH 110 HWK 2: DUE THURSDAY, JULY 3, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2 and 3.A, the Fields and Polynomials supplementary reading handouts, or previously solved
homew
MATH 110 HWK 6: DUE WEDNESDAY, JULY 30, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2, 3, 4, 5, 6, any of the handouts on the website, or previously solved homework problems with
MATH 110 HWK 1 SOLUTIONS
Solve the following problems. Prove all assertions. For each problem, you may use any of the results
in Chapter 1 and 2.A, the Fields supplementary reading handout, or previously solved homework problems
without proof.
1. Let F be
MATH 55 4/21 DISCUSSION QUESTIONS
1.
(a)
(b)
(c)
(d)
(e)
How many relations are there on a set A = cfw_a, b with two elements?
Of these relations, how many are reexive?
How many are symmetric?
How many are transitive?
How many are equivalence relations?
S
MATH 55 4/14 DISCUSSION QUESTIONS
1. If G(x) is the generating function for the sequence cfw_ak , what is the generating function for
each of the following sequences?
(a) 0, 0, 0, a3 , a4 , a5 , .
(b) a0 , 0, a1 , 0, a2 , 0, .
(c) 0, 0, 0, a0 , a1 , a2 ,
MATH 54 FINAL EXAM PRACTICE QUESTIONS
1. Let M be an m n matrix. In terms of the pivots of M , how do we tell if the matrix equation
M x = b always has at least one solution? At most one solution?
2. T/F: In some cases, it is possible for 4 vectors to spa
MATH 54 4/4 QUIZ
There is only one problem on this quiz.
1. Find all dierential functions y satisfying the following conditions:
y 4y + 4y = 0
y (0) = 1
y (1) = 0
Solution: The general solution to the given homogeneous equation is
ygen = Ae2x + Bxe2x .
Pl
MATH 54 4/11 QUIZ
1. Find a general solution to the following dierential equation:
y y = e2t + tet
1
2
MATH 54 4/11 QUIZ
2. Suppose that yp and yq are two solutions to the dierential equation
y + 2y + y = tan3 (x).
Suppose further that
yp (0) = yp (0) = 1
MATH 54 3/7 QUIZ
You have 20 minutes to complete this quiz. No notes or books are allowed.
1.
Let P2 be (as usual) the vector space of polynomials of degree 2. Let D : P2 P2 be the linear
transformation sending p(t) to p (t).
(a) Find the matrix of D rela
MATH 54 3/14 QUIZ
1. Which of the following are orthogonal matrices? Circle all that apply.
0 1/2
a.
0 1/ 2
b.
0 1
1 0
c.
1 2
2 1
d.
1 0
0 1
Solution: Only b and d are orthogonal matrices. A matrix has to have orthonormal columns to be
an orthogonal matri
MATH 54 MIDTERM 1 REVIEW
1. Make sure you review the denitions of the following terms.
(1) augmented matrix vs. coecient matrix
(2) echelon form and reduced echelon form
(3) pivots, pivot rows, pivot columns
(4) parametric form of solution set to Ax = b
(
MATH 54 2/7/2013 QUIZ
You have 15 minutes to complete the following quiz. Notice that there are problems on each
side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more
sophisticated. It really isnt necessary, though.
MATH 54 2/14/2013 DEFINITIONS QUIZ
1. Below is shown a matrix A, together with the reduced
[A|I].
1 4 2 1 0 0
1
[A|I] = 1 3 3 0 1 0 0
3 6 12 0 0 1
0
echelon form of the augmented matrix
0 6 3 4 0
1 1 1 1 0
0 0 3 6 1
Based on this calculation, answer the