Question

triangle abc has vertices a(1,7), b(3,2), and c(-2,-2). triangle abc is rotated 270° clockwise about c. in which quadrant is the image of point a?
a) quadrant i
b) quadrant ii
c) quadrant iii
d) quadrant iv

Explanation:

Step1: Translate point \(A\) relative to \(C\)

First, find the vector from \(C(-2,-2)\) to \(A(1,7)\).
The \(x\)-component: \(1-(-2)=3\), the \(y\)-component: \(7 - (-2)=9\). So the vector is \((3,9)\).

Step2: Apply \(270^{\circ}\) clockwise rotation matrix

The rotation matrix for \(270^{\circ}\) clockwise is \(

$$\begin{pmatrix}0&1\\-1&0\end{pmatrix}$$

\).
Multiply the vector \((3,9)\) by the matrix: \(

$$\begin{pmatrix}0&1\\-1&0\end{pmatrix}$$

$$\begin{pmatrix}3\\9\end{pmatrix}$$

=

$$\begin{pmatrix}9\\-3\end{pmatrix}$$

\)

Step3: Translate back to the original coordinate system

Add the coordinates of \(C(-2,-2)\) to the rotated vector \((9,-3)\).
The \(x\)-coordinate: \(9+(-2) = 7\), the \(y\)-coordinate: \(-3+(-2)=-5\). So the image of \(A\) is \((7,-5)\)

Answer:

D. Quadrant IV