Question

congruence properties of reflections
a. use what you know about perpendicular bisectors and lines of reflections to reflect triangle jkl across the line y = 2.
b. without measuring, how do the sizes of the two triangles compare?

Explanation:

Step1: Recall reflection rule

A reflection across a line is a transformation that flips a figure over the line. If we reflect $\triangle JKL$ across the line $y = 2$, we use the property that for a point $(x,y)$ reflected across the horizontal line $y = k$, the new - point is $(x,2k - y)$. Here $k = 2$.

Step2: Understand congruence property

Reflections are rigid motions. Rigid motions (translations, rotations, and reflections) preserve the size and shape of a figure. That is, the pre - image and the image are congruent. So, when $\triangle JKL$ is reflected across the line $y=2$ to get a new triangle, the two triangles (the original $\triangle JKL$ and its image) are congruent. This means their corresponding side lengths are equal and their corresponding angle measures are equal.

Answer:

a. To reflect $\triangle JKL$ across the line $y = 2$, for each point $(x,y)$ in $\triangle JKL$, the new point $(x',y')$ has coordinates $(x,4 - y)$ since for a reflection across the horizontal line $y = k$, the formula is $(x,2k - y)$ and $k = 2$.
b. The sizes of the two triangles (the original $\triangle JKL$ and its image after reflection across $y = 2$) are equal because reflections are rigid motions that preserve distance (side - lengths) and angle measures. So, the two triangles are congruent.