Question

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rotations

1. given triangle ( abc ) with vertices ( a(-1,0), b(-3,-4) ), and ( c(-4,-1) ), write the

coordinates of its image, triangle ( a^{prime} b^{prime} c^{prime} ), after a rotation of ( 270^{circ} ) clockwise about the origin.
a. ( a^{prime}(1,0), b^{prime}(3,4) ), and ( c^{prime}(4,1) )
b. ( a^{prime}(-1,0), b^{prime}(-3,4) ), and ( c^{prime}(-4,1) )
c. ( a^{prime}(0,1), b^{prime}(4,3) ), and ( c^{prime}(1,4) )
d. ( a^{prime}(0,-1), b^{prime}(4,-3) ), and ( c^{prime}(1,-4) )

Explanation:

Step1: Recall the rotation rule

When a point \((x,y)\) is rotated \(270^{\circ}\) clockwise about the origin, the new coordinates \((x',y')\) follow the rule \((x,y)\to(y, -x)\).

Step2: Apply the rule to point \(A(-1,0)\)

For \(A(-1,0)\), using the rule \((x,y)\to(y, -x)\), we substitute \(x=-1\) and \(y = 0\). Then \(x'=0\) and \(y'=1\), so \(A'=(0,1)\).

Step3: Apply the rule to point \(B(-3, - 4)\)

For \(B(-3,-4)\), substitute \(x=-3\) and \(y=-4\) into the rule \((x,y)\to(y, -x)\). Then \(x'=-4\) and \(y' = 3\), so \(B'=(-4,3)\).

Step4: Apply the rule to point \(C(-4,1)\)

For \(C(-4,1)\), substitute \(x=-4\) and \(y = 1\) into the rule \((x,y)\to(y, -x)\). Then \(x'=1\) and \(y'=4\), so \(C'=(1,4)\).

Answer:

C. \(A'(0,1),B'(-4,3)\), and \(C'(1,4)\)