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 point 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 transformation is $(x,y)\to(x, - y)$. The distance a point moves is twice its $y -$ coordinate value (in magnitude). Let's assume the $y -$ coordinate of point $A$ is $y_A$. The number of units point $A$ moves is $2|y_A|$. For points $B$ and $C$, if their $y -$ coordinates are $y_B$ and $y_C$ respectively, they also move $2|y_B|$ and $2|y_C|$ units.

Step2: Analyze line of reflection

If $\triangle A'B'C'$ is a reflection of $\triangle ABC$, we need to check the line of reflection. If the transformation from $\triangle ABC$ to $\triangle A'B'C'$ follows the rule $(x,y)\to(x, - y)$ for all vertices, the line of reflection is the $x - axis$.

Step3: Determine congruence

Reflections are rigid motions. A rigid motion preserves 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:

(Assuming we fill the drop - down menus):

• To find the reflection of $\triangle ABC$ in the $x - axis$, Point $A$ would move (twice the absolute value of its $y -$ coordinate) units and Points $B$ and $C$ would move (twice the absolute value of their respective $y -$ coordinates) 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 if it is the $x - axis$).