Question

1. ishaan is drawing octagon (q) and its image (q) in the coordinate plane. she wants to be able to identify at least two single - transformations that will map octagon (q) onto octagon (q). draw and label a possible octagon (q) and its image (q) that ishaan could use. what single transformations could you use to map octagon (q) onto octagon (q)? 2. ravi draws rectangle (abcd) in the coordinate plane. he uses a single transformation to form the image rectangle (abcd). a. what transformation could ravi have used? label the vertices of the image to show two different ways ravi could map rectangle (abcd) onto rectangle (abcd) and identify each transformation. b. ravi says there is a third single transformation that maps rectangle (abcd) onto rectangle (abcd). is ravi correct? explain.

Explanation:

Step1: Identify the types of single - transformations

The main types of single - transformations in a coordinate plane are translation, rotation, and reflection.

Step2: For the rectangle ABCD problem part a

• Translation: If we move rectangle ABCD down by a certain number of units to get A'B'C'D', we can label the new vertices accordingly. For example, if we move rectangle ABCD down 5 units, and A=(2,4), B=(4,4), C=(4,2), D=(2,2), then A'=(2, - 1), B'=(4, - 1), C'=(4, - 3), D'=(2, - 3).
• Reflection: If we reflect rectangle ABCD over the x - axis. Let A=(2,4), B=(4,4), C=(4,2), D=(2,2). After reflection over the x - axis, the transformation rule is (x,y)→(x, - y). So A'=(2, - 4), B'=(4, - 4), C'=(4, - 2), D'=(2, - 2).

Step3: For the rectangle ABCD problem part b

The third single - transformation is rotation. If we rotate rectangle ABCD 180° about the origin, the transformation rule is (x,y)→( - x, - y). If A=(2,4), B=(4,4), C=(4,2), D=(2,2), then A'=( - 2, - 4), B'=( - 4, - 4), C'=( - 4, - 2), D'=( - 2, - 2). So Ravi is correct.

Answer:

a. Two ways:

• Translation: Move rectangle ABCD down (or up/left/right) and label new vertices accordingly. For example, if moving down 5 units, if A=(2,4), A'=(2, - 1) etc.
• Reflection: Reflect rectangle ABCD over the x - axis. If A=(2,4), A'=(2, - 4) etc.

b. Ravi is correct. A 180° rotation about the origin is a third single - transformation that can map rectangle ABCD onto A'B'C'D'.