Question

which sequence of transformations maps cdefg onto cdefg? a translation right 8 units and down 14 units followed by a reflection across the line y = -x a translation right 6 units and down 1 unit followed by a reflection across the line y = x a translation right 1 unit and down 7 units followed by a rotation 180° around the origin

Explanation:

Step1: Analyze translation

Let's take a point, say \(C(- 3,8)\). If we consider a translation right \(8\) units and down \(14\) units, the new \(x\) - coordinate is \(-3 + 8=5\) and the new \(y\) - coordinate is \(8-14 = - 6\). After reflection across \(y=-x\), if \((x,y)=(5,-6)\), then the new point is \((6, - 5)\) which is not the correct position of \(C'\).
For a translation right \(6\) units and down \(1\) unit, for point \(C(-3,8)\), the new \(x\) - coordinate is \(-3 + 6=3\) and the new \(y\) - coordinate is \(8 - 1=7\). After reflection across \(y = x\), if \((x,y)=(3,7)\), the new point is \((7,3)\) which is not the correct position of \(C'\).
For a translation right \(1\) unit and down \(7\) units, for point \(C(-3,8)\), the new \(x\) - coordinate is \(-3+1=-2\) and the new \(y\) - coordinate is \(8 - 7 = 1\). A rotation of \(180^{\circ}\) around the origin: If \((x,y)=(-2,1)\), then \((x',y')=(2,-1)\) which is not the correct position of \(C'\).
Let's consider the general rules of transformation. Let's take point \(C(-3,8)\) and \(C'(5,-6)\)
The vector from \(C\) to \(C'\) in \(x\) - direction is \(5-(-3)=8\) and in \(y\) - direction is \(-6 - 8=-14\).
Let's check the reflection across \(y=-x\). The rule for reflection across \(y =-x\) is \((x,y)\to(-y,-x)\)
Let's assume we first translate a general point \((x,y)\) of \(CDEFG\) right \(8\) units and down \(14\) units. The new point is \((x + 8,y-14)\)
Then reflecting \((x + 8,y - 14)\) across \(y=-x\) gives \((-(y - 14),-(x + 8))\)

Answer:

a translation right 8 units and down 14 units followed by a reflection across the line \(y=-x\)