Question

on a math test ali needs to describe the motions which map \\( \triangle efg \\) to \\( \triangle mno \\). the coordinates of \\( \triangle efg \\) are \\( e(5,5) \\), \\( f(7,5) \\), and \\( g(7,10) \\). the coordinates of \\( \triangle mno \\) are \\( m(0,0) \\), \\( n(-2,0) \\), and \\( o(-2,5) \\). he says there is a translation 5 units to the left and 5 units down, followed by a reflection across the \\( x \\)-axis. describe the sequence of rigid motions. what is alis likely error? which of the following is the description of the rigid motions which maps \\( \triangle efg \\) to \\( \triangle mno \\)? a. a translation 5 units to the left and 5 units down, followed by a reflection across the \\( y \\)-axis b. a translation 4 units to the left and 4 units down, followed by a reflection across the \\( y \\)-axis c. a translation 5 units to the left and 5 units down, followed by a reflection across the \\( x \\)-axis

Explanation:

Step1: Analyze translation

For point \(E(5,5)\) to get to a position before reflection: If we consider the \(x\)-coordinate, from \(E(5,5)\) to a point that can be reflected to \(M(0,0)\). First, translation: for \(x\)-coordinate \(5 - 5=0\), for \(y\)-coordinate \(5 - 5 = 0\). After translation \(E(5,5)\to E'(0,0)\), \(F(7,5)\to F'(2,0)\), \(G(7,10)\to G'(2,5)\)

Step2: Analyze reflection

The rule for reflection across the \(y\)-axis is \((x,y)\to(-x,y)\). If we have a point \((x,y)\) and we want to map \(E'(0,0)\to M(0,0)\), \(F'(2,0)\to N(-2,0)\), \(G'(2,5)\to O(-2,5)\) which is a reflection across the \(y\)-axis. If we reflect across the \(x\)-axis the rule is \((x,y)\to(x, - y)\) which would not map \(E'(0,0)\to M(0,0)\), \(F'(2,0)\to N(-2,0)\), \(G'(2,5)\to O(-2,5)\)

Answer:

A. A translation 5 units to the left and 5 units down, followed by a reflection across the \(y\)-axis