geometry question

Question

Choose three non-collinear points on the coordinate plane (make
sure none of your points is the origin). On a sheet of paper, graph
the three points and draw line segments to connect the points and
make a triangle. Label the vertices of the triangle A, B,
and C, and state these vertices in your answer. Now describe
the new coordinates of points A, B, and C
after the following transformations:

a. Translation of point A to the origin.
b. 90° rotation around point B
c. Reflect the triangle across the x-axis.

Answer 1

Choose three points A(4,4),B(6,8),C(4,8) on a graph with origin O(0,0) (O frame), with x the right axis, and y up with positive counterclockwise rotation to form a conventional right hand system. The points can also be considered vectors with associated group properties that we use. A can be translated to O(0,0) by subracting the vector(-4,-4) from the vector(4,4) and then the triangle is A(0,0),B(6,8),C(4,8) in system O. (a). To rotate vectors we can use the Euler transformation matrix with rows[cos(θ),-sin(θ);cos(θ),sin(θ)]} or x′=x*cos(θ)-y*sin(θ), y′=x*sin(θ)+y*sin(θ) or by simply evaluating the new points on the graph in respect to the old using Trigonometry. Now we establish a new coordinate system O¹ with primed origin at B(6,8) in the O frame. They are A¹=(-2,-4),C¹=(2,-4,),B¹=(0,0) in the system with origin at B, called frame O¹. Rotating them 90deg about the origin at B gives the new primed points A²=(4,-2),B²=(0,0),C²=(4,-2) in frame O². We can add the vectors of the O² system to the vector origin of system (O²) measured in the O frame to obtain them in the original system (O frame) or  A=(2,-4),B=(6,8),C'(4,2), (b.). To reflect about the x-axis we can use Euler reflection matrix or {[1, 0;0 -1]} about the x-axis. Then x" = x, y"=-y. The reflected verticies (vectors) are A(4,-4),B(6,-8),C(4,-8) (c.) NOTE: You should check this work as I am not noted for accuracy. Math for general rotation groups is fairly involved but this involves nothing but Algebra and Trionometry and can be done with some effort without much Math.

Answer 2

a)

 

Translation is a movement from one position to another.  This position is a new coordinate point.  The origin is the coordinate (0, 0).

 

(x_A - x_T  ,  y_A - y_T) = (0 , 0)

 

 

We can break this equation into two parts and solve for  x_T and y_T.

 

x_A - x_T = 0               and             y_A - y_T = 0

 

 

(± x_T   ,  ± y_T) is the translation.

 

The plus and minus sign indicates the direction point A takes to get to (0 , 0). 

 

 

b)

 

For each point, find the distance between the point and point B.  That distance will be the same between point B and the newly rotated point.  This is because the point of interest and the rotated point form a 90 degree angle with point B.  Another way to look at it is to draw the point on a coordinate system so that you can get a visual of where the point will end up.  This will help to determine the signs of the x and y coordinates.

 

c)

 

To reflect any point across the x-axis, we only change the signs of the y-coordinate.