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Lesson Explainer: Translations on a Coordinate Plane Mathematics • 11th Grade

In this explainer, we will learn how to translate points, line segments, and shapes on the coordinate plane.

Translations are one of the fundamental ways of moving geometrical objects without changing their shape. In particular, we can recall that translating an object can be thought of as sliding the object in space without changing its size or orientation.

Translations are defined by magnitude (or the distance of the translation) and direction; we can think of this in terms of a horizontal and vertical translation. For example, we can say that a translation moves all objects 1 unit to the right and 2 units up. Let’s see an example of applying this translation to a point 𝐴 with coordinates (3,4). We first plot point 𝐴 on a coordinate grid.

Next, we want to move the point 1 unit to the right and 2 units up.

We call the image of 𝐴 under this translation 𝐴′, and we can find the coordinates of 𝐴′ in two ways. We can read the coordinates of 𝐴′ from the grid to see that 𝐴′(4,6). However, we can also notice that translating 𝐴 1 unit to the right will increase its π‘₯-coordinate by 1 and translating 𝐴 2 units up will increase its 𝑦-coordinate by 2. So, 𝐴′(3+1,4+2)=𝐴′(4,6).

Since a translation slides points the same distance in the same direction, it will affect the coordinates of all points in the same way. This allows us to define translations in the coordinate plane by describing the way they affect the coordinates of a point.

For example, in the above translation that translated 1 unit to the right and 2 units up, we could also say that a point with coordinates (π‘₯,𝑦) would be translated to (π‘₯+1,𝑦+2) since the translation increases the π‘₯-coordinate by 1 and increases the 𝑦-cooridnate by 2. We write this as (π‘₯,𝑦)β†’(π‘₯+1,𝑦+2), where the arrow means β€œis mapped to.”

Let’s now consider the translation (π‘₯,𝑦)β†’(π‘₯βˆ’3,π‘¦βˆ’4) on the point 𝐡(1,2). We calculate that 𝐡′(1βˆ’3,2βˆ’4)=𝐡′(βˆ’2,βˆ’2).

We can also see this translation graphically as shown.

Subtracting 3 from the π‘₯-coordinate translates the point 3 units left, and subtracting 4 from its 𝑦-coordinate translates it 4 units down. Another way of thinking about this is that the horizontal displacement of the translation is βˆ’3 and the vertical displacement is βˆ’4.

We can define this mapping notation formally as follows.

Definition: Translation Mapping in the Coordinate Plane

Any translation in the coordinate plane affects the coordinates of any point in the same way. In particular, it will add or subtract constant values from the π‘₯- and 𝑦-coordinates; these can be thought of as the horizontal and vertical displacement of the translation respectively.

In general, if a translation in the coordinate plane has a horizontal displacement of π‘Ž units and a vertical displacement of 𝑏 units, then (π‘₯,𝑦) will be mapped to (π‘₯+π‘Ž,𝑦+𝑏). We write this as (π‘₯,𝑦)β†’(π‘₯+π‘Ž,𝑦+𝑏). The signs of π‘Ž and 𝑏 tell us the direction of the displacement.

Let’s now see an example of applying this definition to find the coordinates of the image of a point under a given translation.

Example 1: Translating a Point on the Coordinate Plane

Find the coordinates of the image of (13,4) under the translation (π‘₯,𝑦)β†’(π‘₯+5,π‘¦βˆ’2).

Answer

We start by recalling that the notation (π‘₯,𝑦)β†’(π‘₯+5,π‘¦βˆ’2) means that the translation has a horizontal displacement of 5 units and a vertical displacement of βˆ’2 units. In other words, the image will be 5 units to the right and 2 units down. We can find the image of (13,4) by adding 5 to its π‘₯-coordinate and subtracting 2 from its 𝑦-coordinate. We have (13,4)⟢(13+5,4βˆ’2)=(18,2).

This is not the only way of answering this question; we can also see this transformation graphically.

We want to translate the point 5 units to the right and 2 units down. This gives us the point with coordinates (18,2).

In our next example, we will rewrite a translation in terms of horizontal and vertical translations.

Example 2: Identifying Equivalent Translations

Which of the following is equivalent to a translation of (π‘₯,𝑦)β†’(π‘₯+2,π‘¦βˆ’3)?

  1. A translation of 2 units right and 3 units up
  2. A translation of 2 units left and 3 units down
  3. A translation of 2 units right and 3 units down
  4. A translation of 3 units right and 2 units up
  5. A translation of 3 units right and 2 units down

Answer

In the translation (π‘₯,𝑦)β†’(π‘₯+2,π‘¦βˆ’3), we are adding 2 to the π‘₯-coordinate of each point and subtracting 3 from the 𝑦-coordinate. This is equivalent to saying that the translation has a horizontal displacement of 2 and a vertical displacement of βˆ’3 or, equivalently, that the translation is 2 units right and 3 units down.

One way of seeing this is to consider what happens to 𝑂(0,0) under this translation. The translation will add 2 to the π‘₯-coordinate and subtract 3 from the 𝑦-coordinate to give 𝑂′(0+2,0βˆ’3)=𝑂′(2,βˆ’3).

We see that 𝑂′ is 2 units to the right and 3 units down from the origin.

Hence, the answer is option C, a translation of 2 units right and 3 units down.

Before we move onto our next example, we can recall that, in geometric examples, we are often translating along a ray. For example, let’s translate a point 𝐢 𝐴𝐡 units in the direction 𝐴𝐡.

We do this by drawing a ray starting at 𝐢 parallel to 𝐴𝐡 and in the same direction. We then mark the point on this ray that is 𝐴𝐡 units away from 𝐢 as shown. We can construct this parallel ray with a compass and a straightedge.

This allows us to find the image of 𝐢 after the translation; however, we can also think about this translation in terms of horizontal and vertical translations.

By comparing the position of 𝐢 to the position of its image 𝐢′, we see that the translation moves point 𝐢 3 units left and 1 unit down. These are equivalent translations.

This example highlights a few useful properties of translations. First, the size of an object is preserved under translations. In particular, since the size is preserved, the lengths of line segments remain constant under translations. Second, the orientation of a shape is preserved under translations. This means that a line and its image under a translation will remain parallel. Third, translations preserve the measure of any angle. This is a consequence of the lengths being preserved. In particular, triangles are translated onto congruent triangles, so the angle measures stay the same.

Let’s now see an example of translating a triangle in the coordinate plane by using the coordinates of its vertices.

Example 3: Translating a Triangle on the Coordinate Plane given the Magnitude and Direction of the Translation

List the coordinates 𝐴′, 𝐡′, and 𝐢′ that represent the image of triangle △𝐴𝐡𝐢 after translation with magnitude π‘‹π‘Œ in the direction of οƒ«π‘‹π‘Œ, where 𝑋(1,3) and π‘Œ(4,5), given that 𝐴(5,3), 𝐡(1,2), and 𝐢(3,6).

Answer

Let’s start by rewriting the translation in terms of how it affects the π‘₯- and 𝑦-coordinates. To do this, we note that a translation of magnitude π‘‹π‘Œ in the direction of οƒ«π‘‹π‘Œ will map the point 𝑋 to π‘Œ since π‘Œ is π‘‹π‘Œ units away from 𝑋 in the direction οƒ«π‘‹π‘Œ.

We can then determine how the translation affects the π‘₯- and 𝑦-coordinates from 𝑋 to its image π‘Œ.

We see that the π‘₯-coordinate is increased by 3 and the 𝑦-coordinate is increased by 2. So, we can write this translation as (π‘₯,𝑦)β†’(π‘₯+3,𝑦+2).

We can then apply this translation to each vertex of the triangle. We have 𝐴′(5+3,3+2)=𝐴′(8,5),𝐡′(1+3,2+2)=𝐡′(4,4),𝐢′(3+3,6+2)=𝐢′(6,8).

We can check our answer or translate the triangle by plotting the points 𝐴, 𝐡, and 𝐢 together with the image points 𝐴′, 𝐡′, and 𝐢′ on the coordinate plane.

We see that each point is translated 3 units to the right and 2 units up.

Hence, we have shown that 𝐴′(8,5), 𝐡′(4,4), and 𝐢′(6,8).

In our next example, we will find the image of a point on the coordinate plane under a transformation given in terms of its magnitude and direction by a ray between two given points.

Example 4: Finding the Image of a Point after a Translation Given as a Ray between Two Points

The following translation 𝐴𝐡 is equivalent to a horizontal displacement from 1 to 5 and a vertical displacement from 4 to 2. Find the image of point 𝐢 by performing translation 𝐴𝐡 in the direction of 𝐴𝐡.

Answer

We are told that the translation is equivalent to a horizontal displacement from 1 to 5 and a vertical displacement from 4 to 2. We can calculate these displacements. Horizontally, we have 5βˆ’1=4, so the horizontal position increases by 4. Vertically, we have 2βˆ’4=βˆ’2, so the vertical position decreases by 2.

We can write this transformation in mapping notation by noting that the map will increase the π‘₯-coordinate by 4 and decrease the 𝑦-coordinate by 2. Thus, the map is (π‘₯,𝑦)β†’(π‘₯+4,π‘¦βˆ’2). Substituting in the coordinates of 𝐢, we have 𝐢′(1+4,2βˆ’2)=𝐢′(5,0).

We can check that this answer is correct (or find the answer with an alternative method) by applying the translation graphically. Since the translation maps 𝐴 to 𝐡, 𝐢 must be mapped the same distance and direction as 𝐴. So, we can draw a ray starting at 𝐢, which is the same length as 𝐴𝐡 and in the direction of 𝐴𝐡 to find 𝐢′.

By making sure that 𝐴𝐡=𝐢𝐢′ and 𝐢𝐢′ and 𝐴𝐡 have the same direction, we can conclude that 𝐢′(5,0).

In our final example, we will apply a given transformation to three given points on the coordinate plane.

Example 5: Understanding Translations in the Coordinate Plane

Three points (1,βˆ’5), 𝐡(2,βˆ’5), and 𝐢(2,4) are translated by (π‘₯,𝑦)β†’(π‘₯βˆ’3,𝑦+1) to points 𝐴′, 𝐡′, and 𝐢′. Determine 𝐴′, 𝐡′, and 𝐢′.

Answer

We begin by recalling that the notation (π‘₯,𝑦)β†’(π‘₯+π‘Ž,𝑦+𝑏) describes the translation that maps point (π‘₯,𝑦) to point (π‘₯+π‘Ž,𝑦+𝑏). In other words, any point is moved π‘Ž units horizontally and 𝑏 units vertically under this translation.

For the transformation given in the question, we have π‘Ž=βˆ’3 and 𝑏=1, so we are decreasing the π‘₯-coordinate by 3 and increasing the 𝑦-coordinate by 1. We can substitute the π‘₯- and 𝑦-coordinates of each point into the map to determine their images.

We have 𝐴′(1βˆ’3,βˆ’5+1)=𝐴′(βˆ’2,βˆ’4),𝐡′(2βˆ’3,βˆ’5+1)=𝐡′(βˆ’1,βˆ’4),𝐢′(2βˆ’3,4+1)=𝐢′(βˆ’1,5).

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • Any translation in the coordinate plane can be thought of in terms of the horizontal and vertical displacement of the translation.
  • In general, if a translation in the coordinate plane has a horizontal displacement of π‘Ž units and a vertical displacement of 𝑏 units, then (π‘₯,𝑦) will be mapped to (π‘₯+π‘Ž,𝑦+𝑏). We write this as (π‘₯,𝑦)β†’(π‘₯+π‘Ž,𝑦+𝑏).
  • Lengths of line segments are preserved under translations.
  • A line and its image under a translation will be parallel.
  • Angle measure is preserved under translations.

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