Section 3 - Coordinates Flashcards
what idea gives the equation of a line?
that similar △s guarantee that the slope of a line is independent of where it is measured
what is the slope of a line?
m = (y-b)/x where b is the y-intercept
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 ……
y=mx+b
every line has an equation of the form …..
ax+by+c=0
there is a ….. through any two pts
unique line
state the Euclidean distance formula
for any two pts A(x₁,y₁) & B(x₂,y₂), |AB|=√[(x₂-x₁)² +(y₂-y₁)²]
what is the set of all pts equidistant from 2 fixed pts?
a line, this can be proved using coords and the Euclidean distance formula
how can we define a circle?
set of all pts at distance r>0 from a fixed pt O(a,b)
r² = (x-a)²+(y-b)²
a number is constructible from 1 iff …
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
give two lines y=t₁x and y=t₂x what is their relative slope?
± | (t₂-t₁)/(1+t₂t₁) |
what are isometries?
functions that preserve distance
i.e. |PQ|=|f(P)f(Q)|
what are the three main types of isometry
- rotation
- translation
- reflection
prove that a translation is an isometry
- 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) |
how do you rotate the plane about the origin by θ?
send (x,y) to (xcosθ-ysinθ , xsinθ+ycosθ)
the proof that rotation is an isometry is similar to that for translation
how can you reflect about any line?
combine:
- rotation
- translation
- reflection about x-axis