How Do You Find the Center of Rotation in Geometric Transformations?

[DarkPhoenix]

Homework Statement

(1)

A1(2,1) --> B1(3,0)
A2(0,1) --> B2(3,-2)
A3(3,3) --> B3(5,1)

P(4,4)

The reflection, M, Maps Triangle A onto Triangle B
Given that M(P) = Q, write down the coordinates of Q.

C1(-3,4)
C2(-3,2)
C3(-5,5)

The rotation R maps Triangle A onto Triangle C

(i) Find the coordinates of the centre of this rotation
(ii) The angle of direction of this rotation.


Can you explain me in steps how to do this. I don't know how to find centre of Rotation/Reflection.

 

[HallsofIvy]

Homework Statement

(1)

A1(2,1) --> B1(3,0)
A2(0,1) --> B2(3,-2)
A3(3,3) --> B3(5,1)

P(4,4)

The reflection, M, Maps Triangle A onto Triangle B
Given that M(P) = Q, write down the coordinates of Q.

A reflection, in 2 dimensions. Always "reflects" through a straigt line. That means that A1 and B1, A2 and B2, A3 and B3 are on opposite sides of some line and equal distance from that line. That means that the line must pass through the midpoints of the segment from A1 to B1, the segment from A2 to B2, and the segment from A3 to B3.

What are the midpoints of those segments? What is the equation of the line through those midpoints? (A line is determined by 2 points and here you have three midpoints. Find the line through any 2 and, if this really is a reflection, the third midpoint will be on that line.

Once you know that "line of reflection" it should be easy to see what P(4,4) is mapped to.

C1(-3,4)
C2(-3,2)
C3(-5,5)

The rotation R maps Triangle A onto Triangle C

(i) Find the coordinates of the centre of this rotation
(ii) The angle of direction of this rotation.


Can you explain me in steps how to do this. I don't know how to find centre of Rotation/Reflection.

The "perpendicular bisector" of a chord of a circle passes through the center of that circle. Here, A1 and C1, A2 and C2, A3 and C3 are endpoints of chords of circles having the same center. Find the midpoints of segments A1C1, A2C2, and A3C3, find the equations of the perpendicular bisectors (you know how to find the slope of a perpendicular line, don't you?) and find where those three perpendicular bisectors intersect. (Again, it is sufficient to find where 2 of the perpendicular bisectors intersect. If this really is a rotation, the third bisector will intersect in the same point.

 

[Architeuthis Dux]

To find the centre of rotation or reflection, you need to determine the point that stays in the same position before and after the transformation. In this case, we can use the given coordinates of point P (4,4) to find the centre of rotation or reflection.

For reflection, the centre would be the midpoint between the original point and its image. So, the centre in this case would be (3.5, 2.5).

For rotation, the centre would be the point that is equidistant from all three vertices of the triangle. You can find this point by drawing perpendicular bisectors of the sides of the triangle and finding their intersection. In this case, the centre would be (1, 2).

To find the angle of direction for the rotation, you can use the fact that the angle of rotation is equal to the angle between the original point and its image. So, in this case, the angle of direction would be the angle between points P and Q, which you can find using trigonometry or by drawing a right triangle.

In summary, to find the centre of rotation or reflection, you need to use the given coordinates and find the point that stays in the same position before and after the transformation. For rotation, you also need to find the angle of direction, which can be determined by finding the angle between the original point and its image.