How to Determine Transformation Parameters for Triangles in Geometry?

[Johnny Leong]

(a) If a triangle ABC with coordiantes A(2,7), B(2,9) and C(6,7) has a rotation and maps to triangle PQR with coordinates P(6,5), Q(8,5) and R(6,1), what is the centre of rotation? I want to ask in general, what's the way to find the answer?

(b) An enlargement maps the triangle ABC with coordiantes A(2,7), B(2,9) and C(6,7) onto triangle XYZ with coordiantes X(12,12), Y(12,13) and Z(14,12). How to find the centre of enlargement? And this question, the enlargment scale factor is 1/2, right?

(c) A shear maps triangle ABC with coordiantes A(2,7), B(2,9) and C(6,7) onto triangle LMN with coordinates L(2,10), M(2,12) and N(6,16). How to find the shear factor? And is this transformation first with a reflection and then a shear?

(d) A triangle ABC wiht coordinates A(1,1), B(0,2) and C(3,1) is reflected in the line y=-x. How to find the matrix which represents the reflection?

Please help me for these. I need to have examples of this to solve other problems.

 

[ShawnD]

Originally posted by Johnny Leong
(a) If a triangle ABC with coordiantes A(2,7), B(2,9) and C(6,7) has a rotation and maps to triangle PQR with coordinates P(6,5), Q(8,5) and R(6,1), what is the centre of rotation? I want to ask in general, what's the way to find the answer?

If you draw the change in direction for one of the points, you are drawing a secant across the arc that the point actually moved across. The only point where a radius can be drawn perpendicular to the secant as well as the arc is at the midpoint of the secant. Find the midpoint of the secant then draw a line perpendicular to it. This represents the radius for that point.

Do that same process for the other 2 points. Where all 3 radii meet is the centre of rotation.

 

[Johnny Leong]

Originally posted by ShawnD
If you draw the change in direction for one of the points, you are drawing a secant across the arc that the point actually moved across. The only point where a radius can be drawn perpendicular to the secant as well as the arc is at the midpoint of the secant. Find the midpoint of the secant then draw a line perpendicular to it. This represents the radius for that point.

I don't quite understand how to draw the secant and the arc for the change in direction for one of the points. Would you please explain more or have a rough graph for me? Thank you for your attention!

 

[ShawnD]

Sorry if my grammar and wording is hard to understand, I'm just a little tired right now.

Here is a picture of what I mean.

http://myfiles.dyndns.org/pictures/secant.jpg

Draw a line showing the change in position for one of the points on the triangle. Find the midpoint of that change. From the midpoint, draw a perpendicular line.
This 1 line alone will not show you exactly where the centre is but it will point you in the right direction. You will find the centre of rotation by doing this same thing to a different point on the triant then finding where those perpendicular lines drawn intersect.

In my first post I said you need to intersect 3 lines but that's not exactly true now that I think about it. You only need to intersect 2 lines.

 

[HallsofIvy]

You don't really need to "draw" the secant and arc. ShawnD's point was that a rotation moves points in circles. The original point and ending point are on a circle: the secant is simply the line segment connecting them. Knowing the coordinates you can find the midpoint of that line and its slope. The slope of the perpendicular is -1/slope of the line so you can write down its equation. Take another "beginning" point and its "end" point and do the same. The place where they intersect is the center of rotation.

By the way: saying that the rotation takes "the triangle with vertices A B C" into "the triangle with vertices P Q R" does NOT necessarily mean that it maps A into P, B into Q, and C into R. You might want to mark the points on a graph to see which point goes to which.

b) is actually easier: an "enlargement" moves every point out on a "radius"- a line through the center of enlargement. Write the equations of the line through each "beginning" and "ending" point. Solve two of those simultaneous equations for the (x,y) center and check in the third to see that all three line intersect at the same point.

For c) Draw a picture! (Actually you should do that for all of these.) It should be obvious that there is no reflection. Notice that A and B lie on a vertical line A and C lie on a horizontal line. P and Q still lie on a vertical line but P and R do not! What angle is that?

d) Simplest problem of all: Reflecting in the y= -x, (1, 0) changes into what point? (0,1) changes into what point? (DRAW A PICTURE IF YOU ARE NOT SURE!) What matrix does that? Write out a matrix [a b] [1] [0]
[c d] and apply it to [0] and [1].
(I chose (1,0) and (0,1) because they make answering that last question very easy.)

 

[Johnny Leong]

Thank you very much for your nice reply, ShawnD. I have got it. But if the rotation is not 90 degrees, can the same method be applied to find the centre of rotation?

 

[Johnny Leong]

If there's no intersection for the lines joining the points of the original triangle and the corresponding points of the mapped triangle, how can find the centre of enlargement?