Hello BTGmoderatorDC.

Let's take a look at your question.

Let's call R(x,y).

Here, since the point R is in a vertical line that passes through the point Q(5,3), the x-coordinate of R is equal to 5 (x=5).

Now, we just have to find the value of y.

We can do it in two different ways:

1. We can see a right triangle with one angle equal to 45º, so the other angle will be also 45º. So, this triangle is isosceles.

Therefore, the other cathetus is equal to 5, that is to say, y=-5.

So, the coordinates of R are (5,-5).

2. Since the line that passes through r and the origin makes an angle of 45º with the x-axis, then this line is y=-x.

Therefore, the coordinates of R are (x,-x). Since we know x=5 then R(5,-5).

In any case, we get that the correct answer is the option E.

I hope it helps you. <i class="em em-smiley"></i>