Point A is at (3 ,7 ) and point B is at (5 ,-4 ). Point A is rotated pi/2  clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

Jun 1, 2016

(7 ,-3) , ≈ 8.944

Explanation:

The first step is to find the new coordinates of point A , which I will name A'.

Under a clockwise rotation about O of $\frac{\pi}{2}$

a point (x ,y) → (y ,-x)

hence A (3 ,7) → A' (7 ,-3)

To calculate the change in length of AB with A'B use the $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points}$

Length of AB using A(3 ,7) and B(5 ,-4)

d_(AB)=sqrt((5-3)^2+(-4-7)^2)=sqrt125≈11.18

Length of A'B using A'(7 ,-3) and B(5 ,-4)

d_(A'B)=sqrt((5-7)^2+(-4+3)^2)=sqrt5≈2.236

change in length = 11.18 - 2.236 = 8.944