Question

a sequence of transformations is to be performed on \\( \triangle a b c \\). the first part of the sequence of transformations is to rotate the triangle \\( 90^{circ} \\) counterclockwise about the point \\( (0,0) \\).

what are the coordinates of \\( a^{prime}, b^{prime} \\), and \\( c^{prime} \\) ?

a. \\( a^{prime}(2,2), b^{prime}(5,7) \\), and \\( c^{prime}(8,2) \\)
b. \\( a^{prime}(-2,2), b^{prime}(-8,2) \\), and \\( c^{prime}(7,-5) \\)
c. \\( a^{prime}(-2,-2), b^{prime}(-5,-7) \\), and \\( c^{prime}(-8,-2) \\)
d. \\( a^{prime}(-2,2), b^{prime}(-7,5) \\), and \\( c^{prime}(-2,8) \\)

Explanation:

Step1: Recall the rotation rule

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\).

Step2: Find the coordinates of \(A\), \(B\), \(C\)

From the graph, \(A=(2,2)\), \(B=(5,7)\), \(C=(8,2)\).
For point \(A=(2,2)\):
Using the rotation rule \((x = 2,y = 2)\), the new \(x\) - coordinate is \(-y=-2\) and the new \(y\) - coordinate is \(x = 2\), so \(A'=(-2,2)\).
For point \(B=(5,7)\):
Using the rotation rule \((x = 5,y = 7)\), the new \(x\) - coordinate is \(-y=-7\) and the new \(y\) - coordinate is \(x = 5\), so \(B'=(-7,5)\).
For point \(C=(8,2)\):
Using the rotation rule \((x = 8,y = 2)\), the new \(x\) - coordinate is \(-y=-2\) and the new \(y\) - coordinate is \(x = 8\), so \(C'=(-2,8)\).

Answer:

D. \(A'(-2,2)\), \(B'(-7,5)\), and \(C'(-2,8)\)