Question

the graph shows triangles fgh and fgh. which sequence of transformations maps fgh onto fgh? a translation left 1 units and down 7 unit followed by a rotation 90° clockwise around the origin a reflection across the line x = 1 followed by a translation down 7 units a rotation 180° around the origin followed by a reflection across the line x = - 4

Explanation:

Step1: Analyze translation

Let's assume a general point \((x,y)\) in \(\triangle FGH\). A translation left 1 unit and down 7 units would transform the point to \((x - 1,y - 7)\). Then a 90 - degree clock - wise rotation around the origin has the transformation rule \((x,y)\to(y,-x)\). This sequence does not map \(\triangle FGH\) onto \(\triangle F'G'H'\).

Step2: Analyze reflection and translation

For a reflection across the line \(x = 1\), the transformation rule for a point \((x,y)\) is \((2 - x,y)\). Then a translation down 7 units gives \((2 - x,y - 7)\). This sequence does not map \(\triangle FGH\) onto \(\triangle F'G'H'\).

Step3: Analyze rotation and reflection

A 180 - degree rotation around the origin has the rule \((x,y)\to(-x,-y)\). Let's assume a point \((x,y)\) in \(\triangle FGH\), after 180 - degree rotation around the origin, it becomes \((-x,-y)\). Then a reflection across the line \(x=-2\). The rule for reflection across the line \(x = a\) is \((2a - x,y)\), for \(a=-2\), the transformation of \((-x,-y)\) is \((- 4+x,-y)\).
Let's take a vertex of \(\triangle FGH\), say \(F(2,1)\). After 180 - degree rotation around the origin, \(F\) becomes \((-2,-1)\). After reflection across \(x=-2\), the \(x\) - coordinate is \(2\times(-2)-(-2)=-4 + 2=-2\), and the \(y\) - coordinate remains \(-1\).
Let's take \(G(6,2)\). After 180 - degree rotation around the origin, \(G\) becomes \((-6,-2)\). After reflection across \(x=-2\), the \(x\) - coordinate is \(2\times(-2)-(-6)=-4 + 6 = 2\), and the \(y\) - coordinate is \(-2\).
Let's take \(H(7,6)\). After 180 - degree rotation around the origin, \(H\) becomes \((-7,-6)\). After reflection across \(x=-2\), the \(x\) - coordinate is \(2\times(-2)-(-7)=-4 + 7 = 3\), and the \(y\) - coordinate is \(-6\). This sequence maps \(\triangle FGH\) onto \(\triangle F'G'H'\).

Answer:

a rotation 180° around the origin followed by a reflection across the line \(x=-2\)