Question

a triangle has vertices ( a(-2,3) ), ( b(0,0) ), and ( c(1,1) ).
a. find the coordinates of the triangle if it is reflected over the ( x )-axis, then dilated by a scale factor of 3.

Explanation:

Step1: Reflect over the x - axis

When a point \((x,y)\) is reflected over the \(x\) - axis, the transformation rule is \((x,y)\to(x, - y)\).
For point \(A(-2,3)\), after reflection over the \(x\) - axis, it becomes \(A'(-2,-3)\).
For point \(B(0,0)\), after reflection over the \(x\) - axis, it remains \(B'(0,0)\) (since \(y = 0\)).
For point \(C(1,1)\), after reflection over the \(x\) - axis, it becomes \(C'(1,-1)\).

Step2: Dilate by a scale factor of 3

When a point \((x,y)\) is dilated by a scale factor \(k\) centered at the origin, the transformation rule is \((x,y)\to(kx,ky)\). Here \(k = 3\).
For point \(A'(-2,-3)\), after dilation, \(A''=3\times(-2),3\times(-3)=(-6,-9)\).
For point \(B'(0,0)\), after dilation, \(B'' = 3\times0,3\times0=(0,0)\).
For point \(C'(1,-1)\), after dilation, \(C''=3\times1,3\times(-1)=(3,-3)\).

Answer:

The coordinates of the new triangle are \(A''(-6,-9)\), \(B''(0,0)\), and \(C''(3,-3)\)