Question

triangles abc and def are graphed on the set of axes below. which sequence of rigid motions maps △abc onto △def? 1) a reflection over y = -x + 2. 2) a point reflection through (0,2). 3) a translation 2 units left followed by a reflection over the x - axis. 4) a translation 4 units down followed by a reflection over the y - axis.

Explanation:

Step1: Analyze translation

Assume the coordinates of \(A(2,2)\), \(B(7,4)\), \(C(8,1)\) for \(\triangle ABC\).
If we first translate \(\triangle ABC\) \(4\) units down:
The rule for translation \(4\) units down is \((x,y)\to(x,y - 4)\).
For \(A(2,2)\): \((2,2-4)=(2,-2)\)
For \(B(7,4)\): \((7,4 - 4)=(7,0)\)
For \(C(8,1)\): \((8,1-4)=(8,-3)\)

Step2: Analyze reflection

Then reflect the translated points over the \(y\) - axis. The rule for reflection over the \(y\) - axis is \((x,y)\to(-x,y)\)
For \((2,-2)\): \((-2,-2)\) (which is \(F\))
For \((7,0)\): \((-7,0)\) (which is \(D\))
For \((8,-3)\): \((-8,-3)\) (which is \(E\))

Answer:

1. A translation 4 units down followed by a reflection over the \(y\) - axis.