Question

the graph shows triangles efg and efg. which sequence of transformations maps efg onto efg? a rotation 90° clockwise around the origin followed by a reflection across the y - axis a rotation 90° counterclockwise around the origin followed by a translation right 2 units and down 6 units a translation left 6 units and down 3 units followed by a rotation 90° counterclockwise around the origin

Explanation:

Step1: Analyze rotation rules

A 90 - degree clockwise rotation around the origin has the rule $(x,y)\to(y, - x)$. A 90 - degree counter - clockwise rotation around the origin has the rule $(x,y)\to(-y,x)$. A translation of $a$ units left and $b$ units down has the rule $(x,y)\to(x - a,y - b)$. A reflection across the $y$ - axis has the rule $(x,y)\to(-x,y)$.

Step2: Check option 1

For a 90 - degree clockwise rotation around the origin of a point $(x,y)$ in $\triangle EFG$, say $E(3,6)$, it becomes $(6,-3)$. Then reflecting across the $y$ - axis, it becomes $(-6,-3)$. By checking all vertices of $\triangle EFG$ with this sequence, it does not match $\triangle E'F'G'$.

Step3: Check option 2

For a 90 - degree counter - clockwise rotation around the origin of $E(3,6)$, it becomes $(-6,3)$. Then a translation right 2 units and down 6 units: $(-6 + 2,3-6)=(-4,-3)$. By checking all vertices of $\triangle EFG$ with this sequence, it matches $\triangle E'F'G'$.

Step4: Check option 3

For a translation of $E(3,6)$ left 6 units and down 3 units, it becomes $(3 - 6,6 - 3)=(-3,3)$. Then a 90 - degree counter - clockwise rotation around the origin: $(-3,3)\to(-3,-3)$. This does not match the vertices of $\triangle E'F'G'$.

Answer:

a rotation 90° counterclockwise around the origin followed by a translation right 2 units and down 6 units