Question

△abc is shown on the coordinate grid. michael is trying to show that △abc is congruent to △abc by showing that △abc is a reflection of △abc in the x - axis. use the drop - down menus to explain whether michael is correct. to find the reflection of △abc in the x - axis, point a would move choose... units and points b and c choose... units. using the original coordinates, △abc is a reflection of △abc in the line choose... the triangles are choose... the line of reflection michael used is choose...

Explanation:

Step1: Recall reflection rule

When reflecting a point $(x,y)$ over the $x -$axis, the new point is $(x, - y)$. Point $A$ has a non - zero $y$ - coordinate. The distance from point $A$ to the $x$ - axis is equal to its $y$ - coordinate value. If $A=(x_1,y_1)$ where $y_1 = 3$, the distance it moves to its reflection over the $x$ - axis is $2|y_1|=6$ units. Points $B$ and $C$ lie on the $x$ - axis, so they move $0$ units.

Step2: Analyze line of reflection

If we consider the general rule for reflection of a point $(x,y)$ over a horizontal line $y = k$, for reflection over the $x$ - axis ($k = 0$), the transformation is $(x,y)\to(x,-y)$. Looking at the coordinates of the vertices of the triangles, if $\triangle A'B'C'$ is a reflection of $\triangle ABC$, and we assume the correct transformation based on the coordinates, we can see that $\triangle A'B'C'$ is a reflection of $\triangle ABC$ in the $x$ - axis.

Step3: Determine congruence

Reflections are rigid motions. Rigid motions (translations, rotations, and reflections) preserve the size and shape of a figure. So, if $\triangle A'B'C'$ is a reflection of $\triangle ABC$, then $\triangle ABC\cong\triangle A'B'C'$.

Answer:

To find the reflection of $\triangle ABC$ in the $x$-axis, Point $A$ would move $6$ units and Points $B$ and $C$ move $0$ units. Using the original coordinates, $\triangle A'B'C'$ is a reflection of $\triangle ABC$ in the line $x - axis$. The triangles are congruent. The line of reflection Michael used is correct.