Question

pentagon ( defgh ) is the image of pentagon ( defgh ) under a translation followed by a rotation ( 180^{circ} ) about the origin. write the rules for the translation and rotation. translation: ( (x,y)mapsto(square,square) ) rotation: ( (x,y)mapsto(square,square) )

Explanation:

Step1: Determine the translation rule

Take a point, say \(G(7,5)\). After translation (before rotation), assume the translation rule is \((x,y)\to(x + a,y + b)\).
For rotation \(180^{\circ}\) about the origin, the rule is \((x,y)\to(-x,-y)\). Let's first find the translation.
If we consider the relationship between \(G(7,5)\) and \(G'(- 2,-2)\) (after translation and rotation). Let the translation be \((x,y)\to(x + h,y + k)\) and then rotation \((x,y)\to(-x,-y)\).
Let \(G(x = 7,y = 5)\), after translation \(G_1(7 + h,5 + k)\), after rotation \(G'( - (7 + h),-(5 + k))=(-2,-2)\)
We have the system of equations: \(-(7 + h)=-2\) and \(-(5 + k)=-2\)
From \(-(7 + h)=-2\), we get \(7+h = 2\), so \(h=-5\)
From \(-(5 + k)=-2\), we get \(5 + k=2\), so \(k=-3\)
The translation rule is \((x,y)\to(x-5,y - 3)\)

Step2: Determine the rotation rule

The rule for a \(180^{\circ}\) rotation about the origin is a well - known transformation. For any point \((x,y)\) in the coordinate plane, when rotated \(180^{\circ}\) about the origin, the new coordinates \((x',y')\) are given by the formula \((x',y')=(-x,-y)\)

Answer:

Translation: \((x,y)\to(x - 5,y-3)\)
Rotation: \((x,y)\to(-x,-y)\)