Rotation of a triangle

• Mar 29th 2010, 08:40 AM
cozza
Rotation of a triangle
(a) A triangle has vertices at the points A(3,−1), B(−2,1) and C(2,3). Suppose that the triangle is to be moved so that B is at the origin and BA lies along the positive x-axis. One isometry that achieves this transformation is the composite of a translation followed by a rotation. (You may find it helpful to sketch the triangle.)

(i) Determine the translation that moves B to the origin, giving your answer in the form $t_{a,b}$. Write down a formal definition of this translation in two-line notation.

(ii) Find the images A' of A and C' of C under the translation in part (i)

(iii) 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.)

(iv) Find the coordinates of the images of A' and C' under the rotation rθ. Give your answers as exact values.

I have done (i) and (ii) see below, but am unsure about how to work out the rotation without knowing what angle it rotates by. Thanks for your pointers.

(i) $t_{a,b}:R^2 \rightarrow R^2$
$(x,y) \mapsto (x+2,y-1)$

(ii) A' (5,-2) and C' (4,2)

(iii) ?
• Mar 29th 2010, 10:35 AM
qmech
You need to find the rotation matrix in 2 dimensions. See:

Rotation matrix - Wikipedia, the free encyclopedia

for details. All you need is the equation for x' and y' in the section "Dimension 2".

To find the values of cos(theta) and sin(theta), just substitute x for cos(theta) and sqrt(1-x^2) for sin(theta). Then use algebra to force the y component of point A to be zero.
• Apr 5th 2010, 03:13 AM
cozza
Quote:

Originally Posted by cozza
(iii) 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.)

(iv) Find the coordinates of the images of A' and C' under the rotation rθ. Give your answers as exact values.

I have done (i) and (ii) see below, but am unsure about how to work out the rotation without knowing what angle it rotates by. Thanks for your pointers.

(i) $t_{a,b}:R^2 \rightarrow R^2$
$(x,y) \mapsto (x+2,y-1)$

(ii) A' (5,-2) and C' (4,2)

(iii) ?

This question is still driving me crazy! I thought I had it, but obviously not. My tutor added the following comments:

From (i) and (ii) you know the coordinates of A'B'C', so you can plot these. You then need the rotation that makes BA lie along the positive x-axis. If you draw the traingle A'B'C' you will have formed a right-angled traingle and the required theta is one of the angles of the traingle. You can then work out sin, cos and tan of this angle from the triangle. These values can be used in the formal two line definition of a rotation to answer (iii). Then use this rule to find out the images of A' and C'.

We have a software program that by trial and error I worked out the rotation θ = π/8, but I am not sure how to show this. Please help, I am really struggling (Headbang)
• Apr 5th 2010, 10:44 AM
qmech
So you started off with:

A = (3,-1)
B = (-2,1)
C = ( 2,3)

You then translated by -B to get
A' = (5, -2)
B' = (0, 0)
C' = (4,2)

Now you need to rotate by a 2D rotation matrix

$\begin{bmatrix}cos(\theta)&- sin(\theta)\\sin(\theta)&cos(\theta)\end{bmatrix}
\begin{bmatrix}5\\-2\end{bmatrix}=\begin{bmatrix}\sqrt(29)\\0\end{bma trix}$

to get the y-coordinate of A to be zero. Note the sqrt(29) comes from the distance of point A from the origin. Rotations won't change radial distances.

If you solve this for cos(theta), you should get cos(theta) = 5/sqrt(29), if I haven't made any calculation errors. From here you can get sin and tan, etc.

You also use the same rotation matrix on C' to find the image of C' after rotation.