Question

the graph shows triangles lmn and lmn. which sequence of transformations maps lmn onto lmn? a rotation 90° counterclockwise around the origin followed by a reflection across the y - axis a translation up 6 units followed by a rotation 90° counterclockwise around the origin a reflection across the x - axis followed by a translation left 7 units and down 1 unit

Explanation:

Step1: Analyze transformation rules

We know the rules for rotation, reflection and translation. For a 90 - degree counter - clockwise rotation around the origin, the transformation rule for a point $(x,y)$ is $(-y,x)$. For reflection across the $y$-axis, the rule is $(-x,y)$, for reflection across the $x$-axis the rule is $(x, - y)$, and for translation $(x,y)\to(x + a,y + b)$ where $a$ is the horizontal and $b$ is the vertical displacement.

Step2: Check option 1

If we first rotate $\triangle LMN$ 90 - degree counter - clockwise around the origin, and then reflect across the $y$-axis. Let's assume a vertex of $\triangle LMN$ say $L(x_1,y_1)$. After 90 - degree counter - clockwise rotation, it becomes $L_1(-y_1,x_1)$. After reflection across the $y$-axis, it becomes $L_2(y_1,x_1)$. This does not map $\triangle LMN$ to $\triangle L'M'N'$.

Step3: Check option 2

If we first translate $\triangle LMN$ up 6 units. A point $(x,y)$ in $\triangle LMN$ becomes $(x,y + 6)$. Then if we rotate 90 - degree counter - clockwise around the origin, a point $(x,y+6)$ becomes $(-(y + 6),x)$. This maps $\triangle LMN$ to $\triangle L'M'N'$.

Step4: Check option 3

If we first reflect across the $x$-axis, a point $(x,y)$ becomes $(x,-y)$. Then if we translate left 7 units and down 1 unit, a point $(x,-y)$ becomes $(x - 7,-y-1)$. This does not map $\triangle LMN$ to $\triangle L'M'N'$.

Answer:

a translation up 6 units followed by a rotation 90° counterclockwise around the origin