Question

geometry study guide for double transformations
name midolay alizee okeg
period

1. define translation:
2. define rotation
3. define reflection
4. define pre - image
5. define image
6. define line of reflection
7. △abc has vertices a(4, - 2), b(1,1) and c(2, - 3). what are the coordinates of the vertices of △abc after a rotation of 180° clockwise about the origin?

a____ b__ c____

1. describe the transformation and rules that take place in this image
2. the point p(4, - 3) is translated so that (x,y)→(x - 2,y + 3) and then rotated 90° clockwise about the origin. find the coordinates of its image, p.

p = ______

Explanation:

1. Translation: A transformation that moves every point of a figure or a space by the same distance in a given direction. For example, moving a triangle 3 units to the right and 2 units up.
2. Rotation: A transformation in which a figure is turned around a fixed point (the center of rotation). For instance, rotating a square 90 - degrees around its center.
3. Reflection: A transformation that flips a figure over a line (the line of reflection). Like reflecting a rectangle over a vertical line.
4. Pre - image: The original figure before a transformation is applied. For example, the initial triangle before any translation, rotation or reflection.
5. Image: The figure that results from applying a transformation to the pre - image. If a pre - image triangle is rotated, the new triangle is the image.
6. Line of Reflection: The line over which a figure is reflected. It acts like a mirror for the reflection transformation.
7. The rule for a 180 - degree clockwise rotation about the origin is \((x,y)\to(-x,-y)\).

• For point \(A(4, - 2)\), \(A'\) is \((-4,2)\).
• For point \(B(1,1)\), \(B'\) is \((-1,-1)\).
• For point \(C(2,-3)\), \(C'\) is \((-2,3)\).

1. Without seeing the image, a general description could be: If it's a translation, the rule could be \((x,y)\to(x + a,y + b)\) where \(a\) and \(b\) are the horizontal and vertical displacements respectively. If it's a rotation, it could be \(90^{\circ},180^{\circ}\) etc. about a point (usually the origin) with corresponding rules \((x,y)\to(y,-x)\) for \(90^{\circ}\) clockwise rotation about the origin etc. If it's a reflection, it's over a line with rules depending on the line of reflection.
2. First, apply the translation to point \(P(4,-3)\) using the rule \((x,y)\to(x - 2,y + 3)\).

• \(x=4,y = - 3\), so the translated point \(P'\) has coordinates \((4-2,-3 + 3)=(2,0)\).
• Then, apply the 90 - degree clockwise rotation rule \((x,y)\to(y,-x)\) to \(P'(2,0)\). The new point \(P''\) has coordinates \((0,-2)\).

Answer:

1. A transformation that moves every point of a figure or a space by the same distance in a given direction.
2. A transformation in which a figure is turned around a fixed point (the center of rotation).
3. A transformation that flips a figure over a line (the line of reflection).
4. The original figure before a transformation is applied.
5. The figure that results from applying a transformation to the pre - image.
6. The line over which a figure is reflected.
7. A'. \((-4,2)\), B'. \((-1,-1)\), C'. \((-2,3)\)
8. Varies depending on the image (could be translation, rotation or reflection with corresponding rules).
9. \((0,-2)\)