Question

triangle jkl was reflected or rotated and then translated to create triangle jkl. triangles jkl and jkl are shown below. select the sequence that describes the transformation from triangle jkl to triangle jkl. select triangle jkl over the line y = -x, followed by a translation (x, y) → (x + 2, y + 1). rotate triangle jkl 90° ccw about the origin, followed by a translation (x, y) → (x + 2, y + 1). rotate triangle jkl 180° ccw about the origin, followed by a translation (x, y) → (x - 2, y + 1). translate triangle jkl (x, y) → (x + 2, y + 1), followed by a rotation of 180° ccw about the origin.

Explanation:

Step1: Analyze rotation effect

A $90^{\circ}$ clock - wise rotation about the origin transforms a point $(x,y)$ to $(y, - x)$. A $180^{\circ}$ rotation about the origin transforms a point $(x,y)$ to $(-x,-y)$. Reflection over the line $y = - x$ transforms a point $(x,y)$ to $(-y,-x)$.

Step2: Analyze translation effect

A translation $(x,y)\to(x + a,y + b)$ moves the figure $a$ units horizontally and $b$ units vertically.

Step3: Examine each option

For option 1: Reflection over $y=-x$ and then translation. Reflection over $y = - x$ changes the orientation in a specific way. Then adding $(2,1)$ to each coordinate for translation.
For option 2: A $90^{\circ}$ clock - wise rotation about the origin changes the orientation, and then the given translation.
For option 3: A $180^{\circ}$ rotation about the origin flips the triangle across the origin, and then the translation $(x,y)\to(x - 2,y + 1)$.
For option 4: First translation and then rotation. The order of transformation matters.
By observing the orientation and position change of triangle $JKL$ to $J'K'L'$, we can see that rotating triangle $JKL$ $180^{\circ}$ about the origin first (which flips it across the origin) and then translating it according to $(x,y)\to(x - 2,y+1)$ will map triangle $JKL$ to triangle $J'K'L'$.

Answer:

Rotate triangle $JKL$ $180^{\circ}$ about the origin, followed by a translation $(x,y)\to(x - 2,y + 1)$