EXPLORING TRANSLATIONS

Function is a rule that assigns exactly one output to each input.

A transformation is a function that describes a change in the position, size, or shape of a figure. The input of a transformation is the pre-image, and the output of a transformation is the image.

The triangle PQR has the following vertices.

P(1, 11), Q(1, 7) and R(3, 7)

(i)  Trace triangle PQR and line PP′ onto a piece of paper as given below.

(ii) Slide your triangle along the line to model the translation that maps point P to point P′.

(iii)  The image of the translation is the triangle produced by the translation.

Sketch the image of the translation.

(iv) The vertices of the image are labeled using prime notation. For example, the image of P is P′. Label the images of points Q and R.

(v)  Describe the motion modeled by the translation. 

Move 7 units to the right and 5 units down.

(vi)  Check that the motion we described in part (v) is the same motion that maps point Q onto Q′. 

Reflect

Is the orientation of the triangle affected by the translation ? 

No

The orientation of the triangle is not affected and it stays the same.

Practice Problems

Problem 1 :

A triangle has the vertices P (-6, 2), Q(-7, -5) and R(-2, -3).  P', Q'and R' are the vertices of the triangle after a translation of 7 units to the right and 2 units up. Find the vertices P', Q' and R'

Solution : 

Since there is a translation of 7 units to the right and 2 units up, we have to add 7 to x-coordinate and add 2 to y-coordinate of each vertex. 

P' ------>  (-6+7, 2+2)  =  (1, 4)

Q' ------>  (-7+7, -5+2)  =  (0, -3)

R' ------>  (-2+7, -3+2)  =  (5, -1)

Problem 2 :

A trapezoid has the vertices A (1, 0), B(4, 0), C(4, -4) and D(0, -4). A', B', C'and D' are the vertices of the trapezoid after a translation of 5 units to the left and  3 units up. Find the vertices A', B', C' and D'

Solution : 

Since there is a translation of 5 units to the left and 3 units up , we have to subtract 5 from x-coordinate and add 3 to y-coordinate of each vertex. 

A' ------>  (1-5, 0+3)  =  (-4, 3)

B' ------>  (4-5, 0+3)  =  (-1, 3)

C' ------>  (4-5, -4+3)  =  (-1, -1)

D' ------>  (0-5, -4+3)  =  (-5, -1)

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