Question

the diagram shows $\triangle pqr$ after it has been rotated $90^{\circ}$ clockwise. what are the coordinates of $q$?\
\
options: $\\ (7, -4)\\ $, $\\ (4, -7)\\ $, $\\ (-4, 7)\\ $, $\\ (-7, 4)\\ $

Explanation:

Step1: Recall 90° clockwise rotation rule

The rule for rotating a point \((x, y)\) 90° clockwise about the origin is \((x, y) \to (y, -x)\). But wait, actually, if we know the image \(Q'\) and we need to find the pre - image \(Q\) (since \(\triangle P'Q'R'\) is the image after 90° clockwise rotation, so to find \(Q\) we need to reverse the rotation, which is equivalent to rotating \(Q'\) 90° counter - clockwise. The rule for 90° counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\).
Given \(Q'=(4,7)\)

Step2: Apply the reverse rotation rule

For a point \((x,y)\) rotated 90° counter - clockwise, the new coordinates \((x',y')\) are given by \(x'=-y\) and \(y' = x\).
Substitute \(x = 4\) and \(y = 7\) into the formula:
\(x'=-7\) and \(y' = 4\)
So the coordinates of \(Q\) are \((-7,4)\)

Answer:

\((-7,4)\) (corresponding to the option \((-7,4)\))