Question

The verticies of ABC are A(2,4), B (7,6) and C(5,2) give the correct composition of the matricies for the image of ABC after the given composition of the transformations in the order listed(show what it would be for each vertex)

First Translation: (x, y) -> (x-2,y-3)

Second Rotation: 75 degrees about the pivot popoint (1,3)

Answer 1

Step 1.) Translate the given points by (x-2,y-3) Thus after applying translation vertices of ABC will be

A = (2-2,4-3) = (0,1)

B = (7-2,6-3) = (5,3)

C = (5-2,2-3) = (3,-1)

Step 2.) Because we need to rotate around point(1,3). We first need to shift our orgin to point (1,3). To do so we need to subtract 1 and 3 from x and y coordinates respectively.

A = (0-1,1-3) = (-1,-2)

B = (5-1,3-3)=(4,0)

C = (3 - 1,-1 - 3) = (2,-4)

Step 3) Now we will rotate each point 75 degrees usng:

x′=xcosθ−ysinθ

y′=ycosθ+xsinθ

A = x' = -1*cos(75degrees) + 2*sin(75degrees) = 1.67303260748

y' = -2*cos(75) -1*sin(75) = -1.48356391649

B = x' = 4*cos(75degrees) - 0*sin(75degrees) =  1.03527618041

y' = 4*sin(75degrees) = 3.86370330516

C = x' =  2*cos(75degrees) + 4*sin(75degrees) = 4.38134139536

y' = -4*cos(75degrees) + 2*sin(75degrees) = 0.89657547216

STEP 4.) Now shift the origin back to (0,0) by adding 1 and 3 to x and y coordinates respectively.

A = (1.6730+1,-1.4835+3) = (2.67,1.5165)

B = (1.03527618041+1,3.86370330516+3) = (2.03,6.86)

C = ( 4.38134139536+1,0.89657547216+3) = ( 5.38,3.89)