Translation and then a rotation of a triangle
I have a triangle and the vertices were A(3,−1), B(−2,1) and C(2,3). I made a translation on a triangle (in another part of the paper) and now its new
vertices are A(5,−2), B(0,0) and C(4,2).
Next, I must rotate the triangle so that the line BA lies along the positive x-axis.
Here is the questions:
(3) Let rθ be the rotation that completes the required transformation, where θ lies in the interval (−π,π]. Find the exact values of tan θ,cos θand sin θ, and hence write down a formal definition of rθ using two-line notation. (There is no need to work out the value of the angle θ.)
- This is where I have the problems. I worked out as follows, but that is a lot of typing numbers so the answers I found are:
-- Is this correct?
If so, I then tried to make the formal definition and I arrived with (but I can not show the right arrow shape!):
(x,y) |--> (x cos θ - y sin θ, x sin θ + y cos θ)
So I replace all of this with the values at the above paragraph (a lot of typing of the surds, so I don't be putting it all here the working out). We do not show matrices as we are not covering them yet in class, so I must always write with surds.
I finish with the (x,y) values of
(25*sqrt(29)/29 + 45*sqrt(29)/29, 10*sqrt(29)/29 - 10*sqrt(29)/29)
This makes coordinates ( (sqrt(29), 0).
-- But this does not look correct to me. Particular on the drawing I made. Please advise if I am doing this correct.
(4) Find the coordinates of the images of A' and C' under the rotation rθ. Give your answers as exact values. This means surds.
- But I just don't know how to do this question. I wrote a lot of workings and find different answers each time and they don't look right on the drawing that I made of the triangle in the x axis.
-- Can you please help me as I am spending so many hours on this today and yesterday and I go not very far!
(5) Write down a formal definition of the composite transformation: that is the result of the translation followed by the rotation above.
- I can not make this definition without getting the above question 2 well done.
Please help someone who can!