Video Transcript
Determine the coordinates of the
vertices’ images of triangle 𝐴𝐵𝐶 after a counterclockwise rotation of 270 degrees around
the origin.
In this question, we have a rotation,
which means that the shape is going to turn. The point at which it’s going to turn or
the center of rotation is the origin. That’s the coordinate zero, zero. The angle of rotation is 270 degrees and
the direction is counterclockwise. So the angle of 270 degrees will look
like this and the direction will be like this, counterclockwise. This would also be equivalent to a
90-degree clockwise turn. So a 270-degree counterclockwise rotation
will send our triangle into this quadrant.
Let’s begin by looking at our point 𝐴
and establishing how far away it is from our center of rotation. It is negative seven units on the 𝑥-axis
and negative three units on the 𝑦-axis. If we consider this as a 90-degree
clockwise rotation, then it will still be seven units away but this time on the 𝑦-axis and
three units, or rather this time negative three units, on the 𝑥-axis. We can label this new point of the image
as 𝐴 prime. Point 𝐵 is at negative three on the
𝑥-axis and negative four on the 𝑦-axis. And after the rotation, it will be three
units on the 𝑦-axis and negative four on the 𝑥-axis. We can label this as 𝐵 prime and create
the line segment 𝐴 prime 𝐵 prime.
For our final point, 𝐶, we can see that
this is at the coordinate negative six, negative five. So the rotation of this will be at six
units on the 𝑦-axis and negative five on the 𝑥-axis. Once we’ve joined the vertices, it’s
always worth checking that the dimensions of each of the original and the image are the same
size. For example, the line 𝐴𝐶 is two units
down and one unit across. And the line 𝐴 prime 𝐶 prime is also
two units by one unit. We can also compare the lines 𝐵𝐶 and 𝐵
prime 𝐶 prime by noticing that they are both three units by one unit.
When answering this question, we could
also have used the rule that a 270-degree counterclockwise rotation around the origin means
that a point 𝐴 with coordinates 𝑥, 𝑦 will have an image 𝐴 prime with coordinates 𝑦,
negative 𝑥. So if we take our vertex 𝐴 with
coordinates negative seven, negative three, then to find the coordinates of the image 𝐴
prime, the 𝑥-coordinate would be the same as the original 𝑦-coordinate and to find the
𝑦-coordinate, this will be the same as the original 𝑥-coordinate, but with a switched
sign. So here, 𝐴 prime is negative three,
seven. And we can see from our diagram that this
is indeed the coordinate of vertex 𝐴 prime.
In the same way, vertex 𝐵 with
coordinate negative three, negative four becomes the vertex 𝐵 prime where the 𝑥-coordinate
is the original 𝑦-coordinate and the 𝑦-value would be the negative of what was the
original 𝑥-value. And we did indeed find that 𝐵 prime is
at negative four, three. We can also see from using this rule and
also the diagram that 𝐶 prime is at negative five, six. We can then list the coordinates of the
vertices’ image 𝐴 prime, 𝐵 prime, and 𝐶 prime.
