Question

triangle def has coordinates ( d(4,-1) ), ( e(5,2) ), and ( f(1,2) ). determine the coordinates of the vertices of the image after a rotation ( 270^{circ} ) counterclockwise about the origin.

a) ( d(-4,1) ), ( e(-5,-2) ), and ( f(-1,-2) )
b) ( d(1,0) ), ( e(2,3) ), and ( f(-2,3) )
c) ( d(-1,-4) ), ( e(2,-5) ), and ( f(2,-1) )
d) ( d(5,-1) ), ( e(6,2) ), and ( f(2,2) )

Explanation:

Step1: Recall the rotation rule

The rule for a \(270^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(y, - x)\).

Step2: Apply the rule to point \(D(4,-1)\)

For \(D(4,-1)\), using the rule \((x,y)\to(y, - x)\), we substitute \(x = 4\) and \(y=-1\). Then \(D'\) has coordinates \((-1,-4)\) (since \(y=-1\) and \(-x=-4\)).

Step3: Apply the rule to point \(E(5,2)\)

For \(E(5,2)\), substitute \(x = 5\) and \(y = 2\) into the rule \((x,y)\to(y, - x)\). Then \(E'\) has coordinates \((2,-5)\) (since \(y = 2\) and \(-x=-5\)).

Step4: Apply the rule to point \(F(1,2)\)

For \(F(1,2)\), substitute \(x = 1\) and \(y = 2\) into the rule \((x,y)\to(y, - x)\). Then \(F'\) has coordinates \((2,-1)\) (since \(y = 2\) and \(-x=-1\)).

Answer:

C. \(D'(-1,-4)\), \(E'(2,-5)\), and \(F'(2,-1)\)