Question

lesson 13 practice problems

1. here are 2 polygons:

select all sequences of translations,
rotations, and reflections below that
would take polygon p to polygon q.
a. rotate 180° around point a.
b. rotate 60° counterclockwise around point a and then reflect over the line fa.
c. translate so that a is taken to j. then reflect over line ba.
d. reflect over line ba and then translate by directed line segment ba.
e. reflect over the line ba and then rotate 60° counterclockwise around point a.

1. the semaphore alphabet is a way to use flags to signal messages. heres how to signal the letter q. describe a transformation that would take the left hand flag to the right hand flag.

Explanation:

Step1: Analyze option A

Rotating polygon \(P\) 180° around point \(A\) will change the orientation of \(P\) such that it will coincide with polygon \(Q\) as the relative positions of vertices with respect to \(A\) will be reversed in a 180 - degree rotation.

Step2: Analyze option B

A 60° counter - clockwise rotation around point \(A\) will not align the polygon \(P\) in a position where reflecting over line \(FA\) will map it to \(Q\). The angles and positions of vertices after the rotation are not correct for the subsequent reflection to work.

Step3: Analyze option C

Translating so that \(A\) is taken to \(J\) changes the position of \(P\), and then reflecting over line \(BA\) will map the translated polygon to \(Q\) as the reflection will flip the polygon in the correct orientation with respect to \(Q\).

Step4: Analyze option D

Reflecting over line \(BA\) first will change the orientation of \(P\), but translating by directed line segment \(BA\) will not map the reflected polygon to \(Q\). The translation vector is not correct for the reflected polygon to coincide with \(Q\).

Step5: Analyze option E

Reflecting over line \(BA\) changes the orientation of \(P\), and then a 60° counter - clockwise rotation around point \(A\) will not map the polygon to \(Q\) as the combination of these two transformations does not align the vertices of \(P\) with those of \(Q\).

For the second part about the semaphore alphabet, without a grid or more specific reference points, assume the left - hand flag is in a certain position and the right - hand flag is obtained by a 90° counter - clockwise rotation around the person's body (assuming the person is the center of rotation).

Answer:

1. A. Rotate 180° around point \(A\); C. Translate so that \(A\) is taken to \(J\). Then reflect over line \(BA\)
2. A 90° counter - clockwise rotation around the person's body.