Lesson 14

Defining Rotations

  • Let’s rotate shapes precisely.

Problem 1

Draw the image of quadrilateral \(ABCD\) when rotated \(120^\circ\) counterclockwise around the point \(D\).

Quadrilateral A B C D on isometric grid. B aligned vertically above D. Side B C is the longest. Side C D is 2 units long. Sides D A and A B are congruent with obtuse angle in between.


Problem 2

There is an equilateral triangle, \(ABC\), inscribed in a circle with center \(D\). What is the smallest angle you can rotate triangle \(ABC\) around \(D\) so that the image of \(A\) is \(B\)?









Problem 3

Which segment is the image of \(AB\) when rotated \(90^\circ\) counterclockwise around point \(P\)?

Segments on a grid.

Problem 4

The semaphore alphabet is a way to use flags to signal messages. Here's how to signal the letter Q. Describe a transformation that would take the right hand flag to the left hand flag.


Stick-figure person with a square flag in each hand. Left-hand flag has an L. Right-hand flag an R. The right arm extends out horizontally from the body. The left extends upward at an angle.
(From Unit 1, Lesson 13.)

Problem 5

Here are 2 polygons:

Two congruent polygons on isometric grid labeled polygon P and polygon Q.

Select all sequences of translations, rotations, and reflections below that would take polygon \(P\) to polygon \(Q\).


Rotate \(180^\circ\) around point \(A\).


Translate so that \(A\) is taken to \(J\). Then reflect over line \(BA\).


Rotate \(60^\circ\) counterclockwise around point \(A\) and then reflect over the line \(FA\).


Reflect over the line \(BA\) and then rotate \(60^\circ\) counterclockwise around point \(A\).


Reflect over line \(BA\) and then translate by directed line segment \(BA\).

(From Unit 1, Lesson 13.)

Problem 6

  1. Draw the image of figure \(ABC\) when translated by directed line segment \(u\). Label the image of \(A\) as \(A’\), the image of \(B\) as \(B’\), and the image of \(C\) as \(C’\).
  2. Explain why the line containing \(AB\) is parallel to the line containing \(A’B’\).
Figure A B C.
(From Unit 1, Lesson 12.)

Problem 7

There is a sequence of rigid transformations that takes \(A\) to \(A’\), \(B\) to \(B’\), and \(C\) to \(C’\). The same sequence takes \(D\) to \(D’\). Draw and label \(D’\):

Triangles A B C and A prime B prime C prime. Point D is located on side A B.
(From Unit 1, Lesson 10.)