Question

pentagon ( a ^ { prime } b ^ { prime } c ^ { prime } d ^ { prime } e ^ { prime } ) is the image of pentagon ( a b c d e ) under a translation followed by a rotation about the origin.
write the rules for the translation and rotation.
translation: ( ( x , y )
ightarrow ( square , square ) )
rotation: ( ( x , y )
ightarrow ( square , square ) )

Explanation:

Step1: Find the translation rule

First, we need to find a pair of corresponding points before translation. Let's assume the translation is \((x,y)\to(x + a,y + b)\).
Take point \(A(1,-3)\) (before any transformation, assume we first do translation). After translation (before rotation), assume the point is \(A_1(x_1,y_1)\), and after rotation it is \(A'(6,7)\).
Let's first consider the translation part. If we assume the rotation is \(180^{\circ}\) (by observing the orientation change). The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\).
Let's work backward. If after rotation \((x_2,y_2)\to A'(6,7)\), then before rotation (after translation) the point \(A_1(- 6,-7)\).
The original point \(A(1,-3)\). For the translation \((x,y)\to(x + a,y + b)\), we have the equations \(1 + a=-6\) and \(-3 + b=-7\). Solving \(a=-7\) and \(b = - 4\). So the translation rule is \((x,y)\to(x-7,y - 4)\)

Step2: Find the rotation rule

The rotation about the origin. By observing the orientation of the pentagon \(ABCDE\) and \(A'B'C'D'E'\). The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\)

Answer:

Translation: \((x,y)\to(x - 7,y-4)\)
Rotation: \((x,y)\to(-x,-y)\)