Lesson Video: Translations on a Coordinate Plane Mathematics

Video Transcript

In this video, 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 fact, translating an object can be thought of as sliding the object in space without changing its size or its orientation. Translations are defined by two properties: firstly their magnitude, which tells us how far the object needs to move, and secondly their direction, which tells us which way the object is moving. When thinking about the direction of a translation, we describe how far the object has to move horizontally and how far it has to move vertically. For example, we may say that a translation moves an object four units to the right and two units up.

On this coordinate plane, we have a point 𝐴, which has the coordinates three, four. If we move this point one unit to the right and then two units up, it ends up at the point four, six. We call this point the image of 𝐴 after the given translation and denote it using the notation 𝐴 prime. We could also find the image of point 𝐴 following this translation without drawing on a coordinate grid if we consider the effect the translation has on the coordinates of a point.

Moving a point one unit to the right will increase its 𝑥-coordinate by one. And moving a point two units up will increase its 𝑦-coordinate by two. So we could say that under this translation of moving a point one unit right and two units up, the point with coordinates 𝑥, 𝑦 is mapped to the point with coordinates 𝑥 plus one, 𝑦 plus two. So point 𝐴, which had coordinates three, four, is mapped to the point 𝐴 prime, which has coordinates three plus one, four plus two, which is indeed the point with coordinates four, six.

If we want to translate objects to the left or down, then we describe the displacement using negative values. For example, a translation three units to the left would correspond to subtracting three from the 𝑥-coordinate. So the horizontal displacement is negative three. A translation of six units down would correspond to subtracting six from the 𝑦-coordinate. So the vertical displacement is negative six.

We can define translations using mapping notation more formally as follows. If a translation in the coordinate plane has a horizontal displacement of 𝑎 units and a vertical displacement of 𝑏 units, then the point with coordinates 𝑥, 𝑦 will be mapped to the point with coordinates 𝑥 plus 𝑎, 𝑦 plus 𝑏. And we can write this using the mapping notation of an arrow. The signs of 𝑎 and 𝑏 tell us the direction of the displacement: positive for to the right or up and negative for to the left or down. Let’s now consider an example of applying this definition to find the coordinates of the image of a point under a given translation.

Find the coordinates of the image of the point 13, four under the translation 𝑥, 𝑦 is mapped to 𝑥 plus five, 𝑦 minus two.

We’ve been given this transformation algebraically. We can recall that the given mapping notation means that this translation has a horizontal displacement of five units and a vertical displacement of negative two units. In other words, the translation maps each point five units to the right and two units down. It’s two units down because we are subtracting two from the 𝑦-coordinate.

Applying this mapping to the point with coordinates 13, four gives the point with coordinates 13 plus five, four minus two, which is the point with coordinates 18, two. We could also apply this transformation graphically if we had access to squared paper on which to draw a coordinate grid. Here’s the point with coordinates 13, four. To add five to the 𝑥-coordinate, we move five units to the right. And then to subtract two from the 𝑦-coordinate, we move two units down. The image of the point 13, four following this translation is indeed the point with coordinates 18, two.

Let’s now consider another example in which we’ll rewrite a translation in terms of the horizontal and vertical displacements it represents.

Which of the following is equivalent to a translation of 𝑥, 𝑦 is mapped to 𝑥 plus two, 𝑦 minus three? (A) A translation of two units right and three units up. (B) A translation of two units left and three units down. (C) A translation of two units right and three units down. (D) A translation of three units right and two units up. Or (E) a translation of three units right and two units down.

We’ve been given this translation in the form of a mapping, and we want to determine what it means in words. We can recall that, in general, the translation represented by the mapping 𝑥, 𝑦 is mapped to 𝑥 plus 𝑎, 𝑦 plus 𝑏 has a horizontal displacement of 𝑎 units and a vertical displacement of 𝑏 units. The signs of 𝑎 and 𝑏 tell us the direction in which the translation occurs.

Looking at the mapping here, the value of 𝑎 is positive two and the value of 𝑏 is negative three. So the horizontal displacement is positive two, which means two units to the right. The vertical displacement is negative three. And as we are subtracting three from the 𝑦-coordinate, this means three units down. Looking carefully at the five options we were given, we can see that a translation of two units right and three units down is option (C).

Another way of describing translations is as a movement along a ray. For example, let’s consider how we can translate a point 𝐶 𝐴𝐵 units in the direction 𝐴𝐵. Here, the direction of the translation is expressed as the direction of the line connecting 𝐴 to 𝐵. And the magnitude is the length of the line segment 𝐴𝐵 itself. We do this by first drawing in a ray starting at point 𝐶 that is parallel to the line segment 𝐴𝐵 and in the same direction. We then mark on this ray the point that is 𝐴𝐵 units away from 𝐶 so that 𝐴𝐵 is the same length as 𝐶𝐶 prime.

Without squared paper, we can construct this parallel ray using a compass and a ruler. And we can find the position of the image of point 𝐶 on this ray by using compasses with the length set to the length of the line segment 𝐴𝐵. This is one method which allows us to find the image of point 𝐶 following translation. However, we can also think about this translation in terms of the horizontal and vertical displacements.

To travel along the ray from 𝐴 to 𝐵, we need to move three units to the left and one unit down. So, to translate the point 𝐶 𝐴𝐵 units in the direction of the ray 𝐴𝐵, we can also move the point 𝐶 three units to the left and then one unit down. Let’s now consider an example in which we translate a triangle in the coordinate plane by using the coordinates of its three vertices.

List the coordinates 𝐴 prime, 𝐵 prime, and 𝐶 prime that represent the image of triangle 𝐴𝐵𝐶 after translation with magnitude 𝑋𝑌 in the direction of the ray 𝑋𝑌, where 𝑋 has coordinates one, three and 𝑌 has coordinates four, five, given that 𝐴 has coordinates five, three; 𝐵 has coordinates one, two; and 𝐶 has coordinates three, six.

Let’s start by rewriting the translation in terms of how it affects the 𝑋- and 𝑌-coordinates. And to do this, we’ll begin by sketching 𝑋 and 𝑌. A translation of magnitude 𝑋𝑌 in the direction of the ray 𝑋𝑌 is equivalent to the translation that maps the point 𝑋 to the point 𝑌, since 𝑌 is 𝑋𝑌 units away from 𝑋 in the direction of the ray 𝑋𝑌. From our diagram, we can see that this translation will have the effect of increasing the 𝑋-coordinate by three and increasing the 𝑌-coordinate by two. So we can express this translation as a point with coordinates lowercase 𝑥, lowercase 𝑦 is mapped to the point 𝑥 plus three, 𝑦 plus two.

We can then apply this mapping to each vertex of the triangle. For the point 𝐴, which has coordinates five, three, it will be mapped to the point five plus three, three plus two. So 𝐴 prime has coordinates eight, five. The point 𝐵 with coordinates one, two will be mapped to the point with coordinates one plus three, two plus two. So 𝐵 prime has coordinates four, four. Finally, for vertex 𝐶, the point three, six, this is mapped to the point three plus three, six plus two. So 𝐶 prime has coordinates six, eight.

We can check our answer by plotting the points 𝐴, 𝐵, and 𝐶 together with the image points 𝐴 prime, 𝐵 prime, and 𝐶 prime on the coordinate plane. We can see that each point has indeed been translated three units to the right and two units up. So we have that the coordinates of 𝐴 prime are eight, five; the coordinates of 𝐵 prime are four, four; and the coordinates of 𝐶 prime are six, eight.

Looking at the two triangles from the previous example, we can see that they are congruent. That is, they are exactly the same shape and size. This highlights some useful properties of translations. Firstly, 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. So, for example, we could say that the line segment 𝐵𝐶 is the same length as the line segment 𝐵 prime, 𝐶 prime. Secondly, the orientation of a shape is preserved under translation. This means that a line and its image under translation will be parallel. So, for example, the line segment 𝐵𝐴 is parallel to the line segment 𝐵 prime, 𝐴 prime. Thirdly, translations preserve the measure of any angle. So the measure of the angle 𝐵𝐴𝐶 is the same as the measure of the angle 𝐵 prime 𝐴 prime 𝐶 prime. And the same is true for the other two angles in our triangles.

Let’s now consider another example in which we’ll find the image of a point on the coordinate plane under a translation whose magnitude and direction are given in terms of a ray between two given points.

The following translation 𝐴𝐵 is equivalent to a horizontal displacement from one to five and a vertical displacement from four to two. Find the image of point 𝐶 by performing translation 𝐴𝐵 in the direction of the ray 𝐴𝐵.

We’re told that this translation 𝐴𝐵, which is the mapping needed to map point 𝐴 onto point 𝐵, is equivalent to a horizontal displacement from one to five and a vertical displacement from four to two. So we can calculate both the horizontal and vertical displacements. The horizontal position increases by four units, and the vertical position decreases by two units. We can express this translation in mapping notation as the point 𝑥, 𝑦 is mapped to the point 𝑥 plus four, 𝑦 minus two.

From the figure, we can determine that the point 𝐶 has coordinates one, two. So, applying this mapping, point 𝐶 will be mapped to the point with coordinates one plus four, two minus two, which is the point five, zero. We can check that this answer is correct by applying the translation graphically. Since the translation maps 𝐴 to 𝐵, point 𝐶 must be mapped the same distance and direction as point 𝐴. So we can draw a ray starting at point 𝐶, which is the same length as the line segment 𝐴𝐵 and in the direction of the ray 𝐴𝐵, to find the point 𝐶 prime. This confirms that the coordinates of 𝐶 prime, which is the image of point 𝐶 following translation 𝐴𝐵 in the direction of the ray 𝐴𝐵, are five, zero.

We’ll now consider one final example in which we’ll apply a given transformation to three points on the coordinate plane.

Three points — 𝐴 one, negative five; 𝐵 two, negative five; and 𝐶 two, four — are translated by the mapping 𝑥, 𝑦 is mapped to 𝑥 minus three, 𝑦 plus one to points 𝐴 prime, 𝐵 prime, and 𝐶 prime. Determine 𝐴 prime, 𝐵 prime, and 𝐶 prime.

We recall first that the notation 𝑥, 𝑦 is mapped to 𝑥 plus 𝑎, 𝑦 plus 𝑏 describes the translation that maps point 𝑥, 𝑦 to the point with coordinates 𝑥 plus 𝑎 and 𝑦 plus 𝑏. It corresponds to a translation with a horizontal displacement of 𝑎 units and a vertical displacement of 𝑏 units. For the translation in this question, we have 𝑎 equal to negative three and 𝑏 equal to positive one. We are decreasing the 𝑥-coordinate by three and increasing the 𝑦-coordinate by one.

To find the image of each point under this translation, we can substitute the 𝑥- and 𝑦-coordinates of each point into the map. So, for the point 𝐴, which has coordinates one, negative five, its image will be the point with coordinates one minus three, negative five plus one. That’s the point negative two, negative four. 𝐵, which has coordinates two, negative five, will be mapped to the point with coordinates two minus three, negative five plus one. So 𝐵 prime has coordinates negative one, negative four. And finally, point 𝐶, which has coordinates two, four, will be mapped to the point with coordinates two minus three, four plus one, which is the point negative one, five.

So we determined 𝐴 prime, 𝐵 prime, and 𝐶 prime. We could also consider this translation graphically. Here are the points 𝐴, 𝐵, and 𝐶 represented on a coordinate plane. A horizontal displacement of negative three units means we’re decreasing the 𝑥-coordinate of each point by three. And a vertical displacement of positive one unit means we’re increasing the 𝑦-coordinate of each point by one. This confirms that the coordinates of 𝐴 prime, 𝐵 prime, and 𝐶 prime are negative two, negative four; negative one, negative four; and negative one, five, respectively.

Let’s now summarize the key points from this video. Firstly, any translation in the coordinate plane can be thought of in terms of the horizontal and vertical displacements. In general, a translation in the coordinate plane with a horizontal displacement of 𝑎 units and a vertical displacement of 𝑏 units can be expressed as the mapping 𝑥, 𝑦 is mapped to 𝑥 plus 𝑎, 𝑦 plus 𝑏. If 𝑎 is positive, the horizontal displacement is to the right, whereas if 𝑎 is negative, the horizontal displacement is to the left. If 𝑏 is positive, the vertical displacement is up, and if 𝑏 is negative, the vertical displacement is down. We also saw that an object and its image under translation are congruent, which also reveals the following properties. Lengths of line segments are preserved under translations. A line and its image under translation are parallel. And finally, angle measure is preserved under translations.