Find equations for a pair of lines M and N so R(sub n)compose R(sub m) is the given transformation.
A. The translation that maps the origin onto (5,0)
B. The rotation of 180o about a point (Xo, Yo)
C. The rotation of 40o about the origin
D. The translation T with T(x,y) = (x+3, y+4)
To find the equations for a pair of lines that represent a given transformation, we need to understand the properties and effects of each transformation. Let's go through each scenario and find the equations for lines M and N accordingly:
A. The translation that maps the origin onto (5,0):
A translation moves every point in a plane a fixed distance in a specific direction. In this case, we are moving the origin to the point (5,0). To represent a translation, we need to find the vector that represents the distance and direction of the translation.
Let's call line M as the original position of a point in the plane and line N as the translated position of the same point after the transformation. Since we are moving the origin from (0,0) to (5,0), the vector that represents the translation is (5,0) - (0,0) = (5, 0).
Given that the equation of line M is y = m*x + b, where m is the slope and b is the y-intercept, the equation of line N will be y = m*x + b + 0.
Hence, the equations for lines M and N representing the translation are:
M: y = m*x + b
N: y = m*x + (b + 0)
B. The rotation of 180o about a point (Xo, Yo):
A rotation of 180 degrees about a specific point changes the positions of all points in relation to that point. Let's call the given point (Xo, Yo).
To represent a rotation of 180 degrees, we need to find the equation corresponding to the rotation. Given the equation y = m*x + b of line M, we can find line N by applying the rotation formula.
The rotation of a point (x, y) by 180 degrees about the point (Xo, Yo) can be represented by the equations:
x' = 2*Xo - x
y' = 2*Yo - y
To get the equation of line N, we substitute x' for x and y' for y in the equation of line M:
y = m * (2*Xo - x) + b
After simplification, we get:
N: y = -m*x + (2*m*Xo + b)
C. The rotation of 40o about the origin:
For a rotation of 40 degrees about the origin, we can use the rotation formula once again. The equations for a rotation of (x, y) by an angle θ counterclockwise about the origin are:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Using θ = 40 degrees, the equation of line N, rotated from line M, will be:
N: y = (x * cos(40) - y * sin(40)) * m + (x * sin(40) + y * cos(40)) * b
D. The translation T with T(x,y) = (x+3, y+4):
A translation by a vector (a,b) moves every point (x,y) to (x+a, y+b). In this case, we are translating every point by the vector (3,4).
For line M with the equation y = m*x + b, we can find the equation for line N with the translation:
N: y = m*(x+3) + b + 4
= m*x + 3m + b + 4
Therefore, the equations for lines M and N representing the translation are:
M: y = m*x + b
N: y = m*x + (3m + b + 4)
These are the equations for lines M and N that represent the given transformations in the respective scenarios.