Question

ab is rotated about the origin. if the coordinates of a and b are (0,0) and (13,0), respectively, which of the following could not be b?
o b(-6,7)
o b(0,-13)
o b(5,-12)
o b(12,5)

Explanation:

Step1: Recall rotation formula

For a point $(x,y)$ rotated about the origin by an angle $\theta$, the new - coordinates $(x',y')$ are given by $x'=x\cos\theta - y\sin\theta$ and $y'=x\sin\theta + y\cos\theta$. When rotating a point $(x,y)$ about the origin, the distance from the origin $d = \sqrt{x^{2}+y^{2}}$ remains the same. The distance of point $B=(13,0)$ from the origin is $d=\sqrt{13^{2}+0^{2}} = 13$.

Step2: Calculate distances for each option

For option A: If $B'=(12,5)$, then the distance $d_{A}=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13$.
For option B: If $B'=(5,-12)$, then the distance $d_{B}=\sqrt{5^{2}+(-12)^{2}}=\sqrt{25 + 144}=\sqrt{169}=13$.
For option C: If $B'=(0,-13)$, then the distance $d_{C}=\sqrt{0^{2}+(-13)^{2}}=13$.
For option D: If $B'=(-6,7)$, then the distance $d_{D}=\sqrt{(-6)^{2}+7^{2}}=\sqrt{36 + 49}=\sqrt{85}
eq13$.

Answer:

D. $(-6,7)$