Lesson 13

Incorporating Rotations

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Problem 1

Here are 2 polygons:

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

5-sided polygons, P and Q, on an isometric grid. Q is a transformation of P. P has vertices A, B, C, D, and E. Q has vertices J, A, F, G, and H.
A:

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

B:

Rotate \(60^\circ\) 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 \(60^\circ\) counterclockwise around point \(A\).

Solution

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Problem 2

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 left hand flag to the right hand flag.

Q

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.

 

Solution

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Problem 3

Match the directed line segment with the image of Polygon \(P\) being transformed to Polygon \(Q\) by translation by that directed line segment.

Translation 1

Congruent polygons. Polygon Q, translated right of Polygon P. No overlap, no touching edges.

Translation 2

Congruent polygons overlapping. Polygon Q, translated barely up and to the right of Polygon P. Left side of Polygon Q overlaps the right side of Polygon P.

Translation 3

Congruent polygons. Polygon Q, translated above and to the right of Polygon P. Bottom edge of Polygon Q touches the top edge of Polygon P.

Translation 4

Congruent polygons overlapping. Polygon Q, translated down and to the left of Polygon P. Top half of Polygon Q overlaps the bottom half of Polygon P.
A:
A directed line segment, w, slants barely upward and to the right. Endpoint on bottom end, arrow at top end touching a point.
B:
A directed line segment, a, slants upward and to the right. Endpoint on bottom end, arrow at top end touching a point.
C:
A directed line segment, v, slants downward and to the left. Endpoint on top end, arrow at bottom end touching a point.
D:
A horizontal directed line segment, u. Endpoint on left end, arrow at right end touching a point.
1:

Translation 1

2:

Translation 2

3:

Translation 3

4:

Translation 4

Solution

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(From Unit 1, Lesson 12.)

Problem 4

Draw the image of quadrilateral \(ABCD\) when translated by the directed line segment \(v\). Label the image of \(A\) as \(A’\), the image of \(B\) as \(B’\), the image of \(C\) as \(C’\), and the image of \(D\) as \(D’\).

Quadrilateral A B C D and directed line segment, v, on isometric grid.

Solution

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(From Unit 1, Lesson 12.)

Problem 5

Here is a line \(\ell\). 

Plot 2 points, \(A\) and \(B\), which stay in the same place when they are reflected over \(\ell\). Plot 2 other points, \(C\) and \(D\), which move when they are reflected over \(\ell\). 

Line L, with endpoints at the bottom left and top right

Solution

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(From Unit 1, Lesson 11.)

Problem 6

Here are 3 points in the plane. Select all the straightedge and compass constructions needed to locate the point that is the same distance from all 3 points.

Three points, point A to the left of point B, point C above both points A and B.
A:

Construct the bisector of angle \(CAB\).

B:

Construct the bisector of angle \(CBA\).

C:

Construct the perpendicular bisector of \(BC\).

D:

Construct the perpendicular bisector of \(AB\).

E:

Construct a line perpendicular to \(AB\) through point \(C\).

F:

Construct a line perpendicular to \(BC\) through point \(A\).

Solution

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(From Unit 1, Lesson 9.)

Problem 7

This straightedge and compass construction shows quadrilateral \(ABCD\). Is \(ABCD\) a rhombus? Explain how you know.

Constructing a rhombus.

Solution

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(From Unit 1, Lesson 7.)