Question 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 to 𝐴′𝐡′𝐢′. Describe the single transformation that would map 𝐴′𝐡′𝐢′ to 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 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 to 𝐴 prime 𝐡 prime 𝐢 prime. Describe the single transformation that would map 𝐴 prime 𝐡 prime 𝐢 prime to 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 asked about transformations. We can recall that the four different transformations are translation, reflection, rotation, and dilation. So let’s look at the first question where we establish which transformation takes this triangle 𝐴𝐡𝐢 on the left to the triangle 𝐴 prime 𝐡 prime 𝐢 prime. If we look at our list of the four transformations, considering dilation, that’s the transformation that usually changes the size of an object. We can see that our two triangles here are the same size. So therefore, we can rule out dilation. When a translation has been performed, this will keep the object in the same orientation. As our triangles are not the same orientation here, we can rule out translation.

This leaves us with reflection or rotation. It does look as though the shapes could be turned and it could be a rotation. But if we observe the letters and considering 𝐴 and 𝐢, if we were to have rotated this shape, then 𝐢 prime would be left of 𝐴 prime. Therefore, the transformation here is not a rotation. It must be a reflection. We can see that, indeed, our two shapes do look like a mirror image of each other. In order to fully describe a transformation, we couldn’t just, for example, say that it’s a reflection. We’d have to also say the line of reflection.

If we draw a line between each vertex and its image, for example, between 𝐴 and 𝐴 prime, we can see that this line would be two units across and two units up. The halfway point along this line would be one unit across to the right and one unit up. Between the vertices 𝐢 and 𝐢 prime, we can see that it’s five units to the right and five units up. So halfway along would be two and a half units to the right and two and a half units up. Joining these halfway points would give us the line of reflection. This has very helpfully given to us as the line 𝐷𝐸. Putting this into a statement, we would write that this is a reflection in the line 𝐷𝐸. And that’s our answer for the first part of the question.

In the second part of this question, we’re asked to describe the single transformation that would map 𝐴 prime 𝐡 prime 𝐢 prime to 𝐴 double prime 𝐡 double prime 𝐢 double prime. Let’s consider our four transformation options. We can rule out dilation again because these triangles haven’t changed size. And as the triangles are a different way up or a different orientation, we can rule out translation. We can also rule out reflection as these two triangles are not a mirror image of each other. If we take this transformation as a rotation, there are a few things we’ll need to establish.

Firstly, the center of rotation, that’s the point about which the rotation is performed. Next, we’ll also need to find the angle and the direction of the rotation. If we take a look at the vertex, 𝐡 prime, we could see that if we turned it in this direction, it would give us 𝐡 double prime here. This looks as though it could be a 90-degree angle. We’d need to consider where the center of rotation would be however. Let’s try working with this point that’s labeled 𝐹. In this case, a line from 𝐡 prime to this point 𝐹 and then from 𝐹 to 𝐡 double prime would create a 90-degree angle.

If we do the same and check out vertex 𝐴 prime and 𝐴 double prime. In this case, the line between 𝐴 prime and 𝐹 and 𝐹 and 𝐴 double prime would also create a right angle of 90 degrees. We can therefore say that the center of rotation is the point 𝐹 and the angle is 90 degrees. The direction of the rotation would be clockwise. We could therefore give our answer as a complete statement describing this transformation as a rotation of 90 degrees clockwise about point 𝐹. It would also have been correct to give this as a rotation of 270 degrees counterclockwise about point 𝐹.

Let’s look at the final part of this question, are triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime congruent? We can recall that congruent means exactly the same shape and size. All the pairs of corresponding sides and angles in the object and its image would stay the same. The first transformation reflected 𝐴𝐡𝐢 onto 𝐴 prime 𝐡 prime 𝐢 prime. The angles and the side lengths stayed the same, which means that these two triangles are congruent. Then, we rotated 𝐴 prime 𝐡 prime 𝐢 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime. Even though the shape was rotated or turned, all the angles and sides stayed the same, which means that these two triangles are congruent.

Therefore, all three triangles are congruent. So our answer for the final part of this question is yes. Triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime are congruent.

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