Video: Using Transformations to Determine Congruence

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴″𝐡″𝐢″ as seen in the figure. Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′. Describe the single transformation that would map 𝐴′𝐡′𝐢′ onto 𝐴″𝐡″𝐢″. Hence, are triangles 𝐴𝐡𝐢 and 𝐴″𝐡″𝐢″ congruent?

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Video Transcript

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴 prime 𝐡 prime 𝐢 prime which has then been transformed onto triangle 𝐴 double prime 𝐡 double prime 𝐢 double prime as seen in the figure. Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴 prime 𝐡 prime 𝐢 prime. Describe the single transformation that would map 𝐴 prime 𝐡 prime 𝐢 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime. Hence, are triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime congruent?

In this question, we’re told that this triangle 𝐴𝐡𝐢 has been transformed twice. The first transformation will take it to this central triangle, 𝐴 prime 𝐡 prime 𝐢 prime, and the second transformation takes it to the triangle on the right. Let’s think about this first transformation which is the first part of the question. In order to do this, we’ll need to recall the four different transformations. These are translation, reflection, rotation, and dilation. We can think of each of these transformations informally. A translation moves a shape either left or right and up or down. Reflections would create a mirror image. Rotations turn an image about a point. And a dilation would usually change the size of a shape unless the scale factor is one or negative one.

So let’s look at these two triangles. We could immediately rule out the dilation as the triangles are the same size. The triangle 𝐴 prime 𝐡 prime 𝐢 prime isn’t a rotation of the original triangle 𝐴𝐡𝐢 as it hasn’t been turned. And they’re also clearly not a mirror image of each other. So this leaves us with a translation. When we’re describing a translation, we need to say how the shape has moved. This could be in the form of words or in the form of a vector.

When we’re describing the movement, we take each vertex in the original shape and see how it’s moved to the vertex in the image. Taking vertex 𝐴, we can see that this has moved three units to the right and two units upwards. We can check this with another vertex. Looking at point 𝐡 to point 𝐡 prime, we can see that the movement would also be three units to the right and two up. We can therefore give our answer to the first part of the question that this transformation would be a translation three right and two up.

Now, we can look at the second part of the question. This time, we’re asked for the transformation from 𝐴 prime 𝐡 prime 𝐢 prime to 𝐴 double prime 𝐡 double prime 𝐢 double prime. We could look at the list of transformations again. We might immediately notice that these two shapes look like a mirror image of each other, which means that it’s very likely to be a reflection. We can check. It’s not a translation because the two shapes are not in the same orientation. It’s not a rotation as our image has not been turned. And its not a dilation as these two shapes are the same size.

When we are describing a reflection, we need to give the line of reflection. If the line of reflection isn’t obvious, we can find it by joining the vertex of the object and its image. The line of reflection will lie halfway along this. We can draw a dashed line between 𝐢 prime and 𝐢 double prime and notice that the midpoint would lie here. Joining the midpoints would give us the line of reflection. We’ve been very helpfully given these two points 𝐸𝐹, which means that we can define the line of reflection. And so our answer for the second part of the question is that this transformation is a reflection in the line 𝐸𝐹. If we’re given a diagram on a coordinate grid, we can also define the line of reflection by using the equation of the line.

Let’s look at the final question which asks if triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime are congruent. That was our first triangle on the left and then the triangle on the right. We can remember that congruent means the same shape and size. More mathematically, we can say that all the corresponding pairs of angles are equal and all the corresponding pairs of sides are equal. The first transformation that we had was a translation. As this really means we’re just moving the shape, then the size wouldn’t change at all. When all the angles stay the same and all the sides stay the same, then we know that a translation will always produce a congruent image. We therefore can say that 𝐴𝐡𝐢 and 𝐴 prime 𝐡 prime 𝐢 prime are congruent.

But what about the second transformation? In this case, we had a reflection. The shapes are now a mirror image of each other, but we still have a congruent shape as all the corresponding sides are the same and all the corresponding angles are the same. In a reflection, we will always have a congruent object and its image. This means that 𝐴𝐡𝐢 will be congruent with 𝐴 double prime 𝐡 double prime 𝐢 double prime. Therefore, the answer for the third part of the question is yes, these triangles are congruent.

As an aside, in a rotation, we would also have a congruent object and image. But in a dilation, we would get similar shapes rather than congruent shapes. Similar shapes are the same shape, but a different size. And that will be because, in a dilation, the shape will have a scale factor. The image will either be larger or smaller than the original shape unless, of course, the scale factor is one or negative one, in which case we would have a congruent shapes.

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