Reflecting and Translating Points

Question

Point R (-4,5) is reflected about the x-axis onto point R'. It is then reflected about the line y=-x to find R''. It is then translated using the vector <2,-2> to find R'''. Find R'''

Answer 1

We use the notation $H_a$ to be the half-turn (or reflection) of a point around an arbitrary line $a$. If you don't know the formulae you should be able to deduce them quite easily (you're encouraged to do so in any case).

• We start with $R(-4,5)$. We know that $H_{y=0}(x,y)=(x,-y)$. Therefore, $R'=H_{y=0}(R)=H_{y=0}(-4,5)=(-4,-5)$. So $R' = (-4,-5)$.

• Now we apply $H_{y=-x}$ to $R'$. It's not hard to convince yourself that $H_{y=-x}(a,b)=(-b,-a)$ (you can also prove it using the formula for the distance between a point and a line). Therefore, $R''=H_{y=-x}(R')=H_{y=-x}(-4,-5)=(5,4)$. So $R''=(5,4)$.

• Finally, we have to translate $R''(5,4)$ by the vector $\langle 2,-2\rangle$. We just add the coordinates and therefore our point $R'''$ is equal to $(7,2)$.

Answer 2

If you have no formulas for this kund of problem, grab a sheet of graph paper and perform the operations graphically. The coordinates are all so simple that you will be able to find the exact result from this.