Question Video: Describing the Translation of the Vertices of a Triangl

A triangle with vertices (3, 3), (7, 0), and (10, 5) was transformed to (1, 8), (5, 5), and (8, 10) and then to (1, 8), (−2, 4), and (3, 1). Which of the following describes these transformations? [A] It was translated 2 units right and 5 units down, and then it was rotated 90° counterclockwise about the point (1, 8). [B] It was translated 2 units left and 5 units up, and then it was rotated 180° clockwise about the point (1, 8). [C] it was rotated 180° counterclockwise about the point (3, 3), and then it was translated 2 units right and 5 units down. [D] It was translated 2 units left and 5 units up and then it was rotated 90° clockwise about the point (1, 8).

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Video Transcript

A triangle with vertices three, three; seven, zero; and 10, five was transformed to one, eight; five, five; and eight, 10 and then to one, eight; negative two, four; and three, one. Which of the following describes these transformations?

And we’re given four answer options. It will be very difficult to answer this question without drawing out the coordinates. So, let’s get some grid paper and draw the vertices of the original triangle. When we’re planning our graph, it can be very helpful to make sure that we have enough room to cover the highest and lowest of our 𝑥- and 𝑦-values. Here, we have our first three coordinates drawn for the original triangle. We can then draw the second set of three coordinates to find the image of the triangle, which will look like this.

We now need to work out which transformation will take the original blue triangle to the triangle drawn in pink. We can recall that the four types of transformations are translation, reflection, rotation, and dilation. Looking at our two triangles, we can see that these are the same size. So, we can eliminate dilation as a transformation option. In a dilation, the shape and its image are usually a different size. In the diagram, we can see that the two triangles are in the same orientation. In other words, they’re not a reflection or mirror image, and they’re not a rotation of each other.

This leaves us with translation, which is when we simply move a shape to the right or left and up or down. In order to work out how a shape has been translated, we compare the vertices. Looking at the top vertex of each triangle and beginning with the blue triangle, we could see that there is a movement of two units to the left and five units upwards. We must be very careful to work with the original triangle and the image. Otherwise, instead of correctly describing this as a translation two units to the left and five units upwards, we could incorrectly describe it as a translation of two units to the right and five units down. But we can make a note that the first transformation will be a translation of two units left and five units up.

We can now look at the final set of three coordinates to find the second transformation. Plotting the coordinates one, eight; negative two, four; and three, one will give us a triangle that looks like this. We need to work out the transformation that takes our triangle in pink to the triangle in orange. Looking at our four transformation options, we can eliminate dilation as these triangles are the same size. We can also eliminate another translation as these two triangles are in a different orientation. The triangles are not a reflected or mirror image of each other. So we can eliminate reflection, which leaves us with rotation.

In order to describe a rotation, we need to give the center of rotation, the angle, and the direction. Sometimes, it can be difficult to find the center of rotation. What we’re really looking for is a point about which the shapes can pivot or turn. If we were to draw the pink triangle on tracing paper and then put the point of our pencil on this green 𝑥. Then moving our tracing paper in the direction of the green arrow would take us to the orange triangle. This is the clockwise direction.

In order to find the angle of rotation, we look at the angle between each vertex and its image. Here, we can see that we have a right angle. So, the angle of rotation will be 90 degrees. Putting these pieces of information together — where the center of rotation is at one, eight, the angle is 90 degrees, and the direction is clockwise — we would have a rotation of 90 degrees clockwise about one, eight. We could also describe this transformation as a rotation of 270 degrees counterclockwise about one, eight.

So if we look at our answer options, the first transformation was a translation of two units left and five units up. So, we can eliminate options (A) and (C). Our second transformation is a rotation of 90 degrees clockwise. So, we can eliminate option (B) as that has a rotation of 180 degrees. We can see the fully correct answer is given in option (D). It was translated two units left and five units up. And then it was rotated 90 degrees clockwise about the point one, eight.

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