Video Transcript
What is the image of the point
negative three, 14 after a rotation about the origin through an angle of 90
degrees?
In this question, we are
considering the rotation of a single point. In order to perform a rotation, we
need to know several important facts: the center of rotation, the angle of rotation,
and the direction of rotation. Here, the origin is the center of
rotation. That’s the coordinates zero,
zero. And the angle of rotation is 90
degrees. However, we aren’t explicitly given
a direction. So that means we will apply the
convention that if the angle is positive, as it is here, then the direction of
rotation is in the counterclockwise direction.
Now, the most efficient way to find
the image of this point is to apply one of the properties of rotation about the
origin: this one in particular, that a rotation of 90 degrees counterclockwise about
the origin is equivalent to the coordinate transformation that the coordinates 𝑥,
𝑦 map to the coordinates negative 𝑦, 𝑥. So we take the coordinates of
negative three, 14. We switch the 𝑥- and
𝑦-coordinates. And then we take the negative of
the old 𝑦-coordinate, which is now in the position of the new 𝑥-coordinate. So the value of 14 is now negative
14. Notice that if the original
𝑦-coordinate was already negative, then it would change to be a positive value.
And that’s the answer for the image
of the point negative three, 14 after a rotation of 90 degrees about the origin. It’s the coordinates negative 14,
negative three.
But of course, it’s always
worthwhile checking our answer. Even a very quick sketch can be
useful. The original point, negative three,
14, would lie here in the second quadrant. So a rotation of 90 degrees
counterclockwise would send this point into the third quadrant. In this quadrant, both coordinates
will be negative. And we know that they can’t be the
other way round because the point negative three, negative 14 wouldn’t be a rotation
of 90 degrees counterclockwise. In fact, it looks more like a
reflection in the 𝑥-axis, which isn’t what we need here. Therefore, the answer for the image
of the point negative three, 14 after the given rotation about the origin is the
point negative 14, negative three.
