Section 3 - Coordinates Flashcards

1
Q

what idea gives the equation of a line?

A

that similar △s guarantee that the slope of a line is independent of where it is measured

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2
Q

what is the slope of a line?

A

m = (y-b)/x where b is the y-intercept

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3
Q

if L is a line that crosses the y-axis at Q=(0,b) and has slope m, then (x,y) is on L iff ……

A

y=mx+b

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4
Q

every line has an equation of the form …..

A

ax+by+c=0

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5
Q

there is a ….. through any two pts

A

unique line

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6
Q

state the Euclidean distance formula

A

for any two pts A(x₁,y₁) & B(x₂,y₂), |AB|=√[(x₂-x₁)² +(y₂-y₁)²]

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7
Q

what is the set of all pts equidistant from 2 fixed pts?

A

a line, this can be proved using coords and the Euclidean distance formula

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8
Q

how can we define a circle?

A

set of all pts at distance r>0 from a fixed pt O(a,b)

r² = (x-a)²+(y-b)²

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9
Q

a number is constructible from 1 iff …

A

it can be obtained from 1 by +, -, *, /, √

this gives us that we cannot construct a square with the same area as a circle, we cannot duplicate the cube with side a, and we cannot trisect all angles

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10
Q

give two lines y=t₁x and y=t₂x what is their relative slope?

A

± | (t₂-t₁)/(1+t₂t₁) |

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11
Q

what are isometries?

A

functions that preserve distance

i.e. |PQ|=|f(P)f(Q)|

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12
Q

what are the three main types of isometry

A
  • rotation
  • translation
  • reflection
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13
Q

prove that a translation is an isometry

A
  • let (x₁,y₁) & (x₂,y₂) be reals
  • then | tₘ,ₙ((x₁,y₁))tₘ,ₙ((x₂,y₂)) | = | (x₁+m , y₁+n)(x₂+m , y₂+n) | = √[(x₁+m-x₂-m)²+(y₁+n-y₂-n)²] = | (x₁,y₁)(x₂,y₂) |
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14
Q

how do you rotate the plane about the origin by θ?

A

send (x,y) to (xcosθ-ysinθ , xsinθ+ycosθ)

the proof that rotation is an isometry is similar to that for translation

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15
Q

how can you reflect about any line?

A

combine:

  • rotation
  • translation
  • reflection about x-axis
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16
Q

prove that if f & g are isometries, then so is f◦g (and g◦f)

A
  • take 2 pts (x,y) & (a,b)
  • then | f◦g((x,y)) f◦g((a,b)) | = | f(g((x,y))) f(g((a,b))) |
  • = | g((x,y)) g((a,b)) | since f is an isometry
  • = | (x,y) (a,b) | since g is an isometry
17
Q

what is a glide reflection?

A

a reflection followed by a translation in the “direction” of the line of reflection

they are NOT translations, rotations or reflections

18
Q

state the three reflections theorem

A

any isometry f of ℝ² is a composition of 1, 2, or 3 reflections