Lesson 12

Defining Translations

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Problem 1

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 P and Q. Polygon Q is located horizontally to the right of Polygon Q. There is a gap between the polygons; they do not overlap.

Translation 2

Congruent polygons P and Q. Polygon Q is located horizontally to the right of Polygon P. The polygons overlap.

Translation 3

Congruent polygons P and Q. Polygon Q is located up and to the left of Polygon P.

Translation 4

Congruent polygons P and Q. Polygon Q is located down and to the right of Polygon P.
A:
Directed line segment, u, pointing horizontally to the right. The directed line segment is the longest of the choices.
B:
Directed line segment, w, pointing down and to the right.
C:
Directed line segment, a, pointing up and to the left.
D:
Directed line segment, v, pointing horizontally to the right. The directed line segment is the shortest of the choices.
1:

Translation 1

2:

Translation 2

3:

Translation 3

4:

Translation 4

Solution

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Problem 2

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 on an isometric grid. B is aligned vertically above D. Side C D is 2 units long. Directed line segment, v, points up and to the right, but more right than up.

Solution

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Problem 3

Which statement is true about a translation?

A:

A translation takes a line to a parallel line or itself.

B:

A translation takes a line to a perpendicular line.

C:

A translation requires a center of translation.

D:

A translation requires a line of translation.

Solution

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Problem 4

Select all the points that stay in the same location after being reflected across line \(\ell\). 

Line L, facing upward and to the right, goes through Points C and E. Points A, B and D sit to the left of the line.
A:

A

B:

B

C:

C

D:

D

E:

E

Solution

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(From Unit 1, Lesson 11.)

Problem 5

Lines \(\ell\) and \(m\) are perpendicular. A point \(Q\) has this property: rotating \(Q\) 180 degrees using center \(P\) has the same effect as reflecting \(Q\) over line \(m\). 

\(m \perp \ell\)

Perpendicular lines L and M, intersecting at point P. L is horizontal, m is vertical.
  1. Give two possible locations of \(Q\).
  2. Do all points in the plane have this property?

Solution

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(From Unit 1, Lesson 11.)

Problem 6

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 B C.

Solution

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(From Unit 1, Lesson 10.)

Problem 7

Two distinct lines, \(\ell\) and \(m\), are each perpendicular to the same line \(n\). 

  1. What is the measure of the angle where line \(\ell\) meets line \(n\)?
  2. What is the measure of the angle where line \(m\) meets line \(n\)?

Solution

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(From Unit 1, Lesson 6.)