Question

△abc with vertices a(-4,6), b(-8,6), and c(-2,8) is dilated with a scale factor of 0.5, then rotated 270° about the origin. determine which quadrant the image is located in.
quadrant i
quadrant ii
quadrant iii
quadrant iv

Explanation:

Step1: Find the coordinates after dilation

When a point \((x,y)\) is dilated with a scale factor \(k = 0.5\) about the origin, the new coordinates \((x',y')\) are given by \(x'=0.5x\) and \(y' = 0.5y\).
For \(A(-4,6)\): \(A'=(0.5\times(-4),0.5\times6)=(-2,3)\)
For \(B(-8,6)\): \(B'=(0.5\times(-8),0.5\times6)=(-4,3)\)
For \(C(-2,8)\): \(C'=(0.5\times(-2),0.5\times8)=(-1,4)\)

Step2: Find the coordinates after rotation

When a point \((x,y)\) is rotated \(270^{\circ}\) counter - clockwise about the origin, the transformation rule is \((x,y)\to(y, - x)\)
For \(A'(-2,3)\): \(A''=(3,2)\)
For \(B'(-4,3)\): \(B''=(3,4)\)
For \(C'(-1,4)\): \(C''=(4,1)\)

Answer:

Quadrant I