﻿ Translation Vectors Worksheet | Problems & Solutions # Translation Vectors Worksheet

Translation Vectors Worksheet
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1.
Which solid can be represented as a three dimensional figure using the given figure and vector?  a. cuboid b. trapezoid c. cone d. rectangle

#### Solution:

A square is given. So, a rectangular prism can be formed from it.

A translation vector indicating 2 units right and 2 units down is given.

Translate each vertex 2 units to the right and 2 units down. Connect the points to form a three dimensional figure.

The figure represents a cuboid.

2.
The figure M'N'O'P' is the image of MNOP. Describe the translation vector by using ordered pair notation.  a. <-14, 8> b. <8, 8> c. <- 14, 14> d. <-8, 8>

#### Solution:

Identify the coordinates of the MNOP and M'N'O'P'.

The coordinates of MNOP are M(- 3, 3), N(- 1, 5), O(6, 4) and P(2, - 2).

The coordinates of M'N'O'P' are M'(- 11, 11), N'(- 9, 13), O'(- 2, 12) and P'(1, - 6).

Coordinates of image - Coordinates of preimage = Coordinates of the translation vector.

The translation that maps M to M' is (-11 - (- 3), 11 - 3) = (- 8, 8)

Horizontal translation in left direction results in negative x-coordinate and translation in right direction results in positive x-coordinate.

Vertical translation in upward direction results in positive y-coordinate and translation in downward direction results in negative y-coordinate.

The translation is 8 units left and 8 units up.

Hence the translation vector that maps PQRS to P'Q'R'S' is <- 8, 8>.

3.
Identify the pattern which is not created by translation.  a. Figure 4 b. Figure 1 c. Figure 3 d. Figure 2

#### Solution:

A translation is a transformation that moves points the same distance and in the same direction.

The pattern 1 is formed by moving a single figure the same distance and in the same direction. So, it is a translation.

The pattern 2 is also formed by moving a single figure the same distance and in the same direction. So, it is a translation.

The pattern 3 is formed by rotating a single figure. So, it is not a pattern created by translation.

The pattern 4 is also formed by moving a single figure the same distance and in the same direction. So, it is a translation.

Hence pattern in figure 3 is not created by translation.

4.
The figure M'N'O'P' is the image of MNOP. Describe the vector by using ordered pair notation.  a. <7, 7> b. <- 3, - 7> c. <7, 5> d. <3, 7>

#### Solution:

The coordinates of M are (5, 6).

The coordinates of M' are (2, - 1).

The movement in left direction results in negative x-coordinate and the movement in right direction results in positive x-coordinate.

The movement in downward direction results in negative y-coordinate and the movement in upward direction results in positive y-coordinate.

Coordinates of preimage + Coordinates of translation = Coordinates of the image. The translation vector is <2 - 5, - 1 - 6> = < - 3, - 7>
[Coordinates of image - Coordinates of preimage = Coordinates of the translation vector.]

Hence the translation vector is < - 3, - 7>.

5.
Which solid can be represented as a three dimensional picture using the figure and vector?  a. pyramid b. sphere c. cylinder d. circle

#### Solution:

A circle is given. So, a cylinder can be formed from it.

Translate each vertex 1 unit to the right and 5 units down. Connect the points to form a cylinder.

6.
Find the single transformation that has the same effect as composition of translations. <3, 7> followed by <11, - 6> followed by <- 8, 3> a. <5, 11> b. <7, 4> c. <-6, -4> d. <6, 4>

#### Solution:

A translation is a transformation that moves points the same distance and in the same direction.

The first translation is <3, 7>.

The second translation is <11, - 6>.

The third translation is <- 8, 3>.

The single transformation which is equivalent to the three translations obtained by adding all the three translations.

The single transformation is <3 + 11 - 8, 7 - 6 + 3> = <6, 4>

7.
Find the image of A after a translation of <- 2, - 6>.  a. B b. E c. D d. C

#### Solution:

The coordinates of A are (4, 4).

The translation vector is <- 2, - 6>.

Coordinates of preimage + Coordinates of translation = Coordinates of the image

The coordinates of A after translation are (4 + (- 2), 4 + (- 6)) = (2, -2)

Hence the image of circle A after translation is represented in circle D.

8.
On what does a vector quantity depend? a. momentum and isolation of a vector b. isolation of vector c. momentum of vector d. distance and direction

#### Solution:

The distance and direction of a translation is expressed as a vector.

Hence the vector quantity depends on the distance and the direction.

9.
Identify the incorrect statement.
I. A translation is a transformation that moves points the same distance and in the same direction
II. A translation is an isometry
III. A translation changes the shape and the size of the figure
IV. A translation does not change orientation a. III b. II c. IV d. I

#### Solution:

A translation is a transformation that moves points the same distance and in the same direction.

A translation is an isometry.

A translation does not change shape and size of the figure.

A translation does not change orientation.

Hence the statement III is incorrect.

10.
Find the image of the polygon ABCD under the translation of <0, -5> followed by <-1, -2> using matrix addition.  a. MNOP b. GFEH c. KJIL d. SRQT

#### Solution:

Identify the vertices of the polygon ABCD.

The vertices of the pre-image are A(1, 3), B(3, 2), C(4, 3) and D(3, 4).

The vertices of the pre-image in matrix form
 A B C D (1 3 4 33 2 3 4)

The first translation vector is <0, -5>.

The translation vector in matrix form is (0 0 0 0-5 -5 -5 -5)

Vertices of the image = Vertices of pre-image + Translation matrix.

Vertices of the image = (1 3 4 33 2 3 4) + (0 0 0 0-5 -5 -5 -5) = (1 3 4 3-2 -3 -2 -1)

The vertices of the polygon after first translation in matrix form is
 A′ B′ C′ D′ (1 3 4 3-2 -3 -2 -1)

The vertices of the image are (1 3 4 3-2 -3 -2 -1) + (-1 -1 -1 -1-2 -2 -2 -2) = (0 2 3 2-4 -5 -4 -3)

The vertices of the image after translations <0, -5> and <-1, -2> are
 A B C D (0 2 3 2-4 -5 -4 -3)