Question

the image of δabc after a reflection across $overleftrightarrow{ec}$ is δabc. which triangle must be a right triangle and why?
○ δabc is right because it is the image of δabc.
○ δadc is right because $overline{aa}$ intersects $overline{ac}$ at a.
○ δbcc is right because b lies of the line of reflection.
○ δbgc is right because $overleftrightarrow{ec} perp overline{cc}$.

Explanation:

1. Analyze Option 1: A reflection preserves the shape's properties, but just being an image doesn't make it right. $\triangle ABC$ isn't stated as right, so $\triangle A'B'C'$ isn't necessarily right. Eliminate.
2. Analyze Option 2: Intersection at $A$ doesn't imply a right angle. $\overline{AA'}$ and $\overline{AC}$ intersecting at $A$ doesn't give a right angle. Eliminate.
3. Analyze Option 3: $B$ on the reflection line means $B = B'$, but that doesn't make $\triangle BCC'$ right. No info shows $\angle BCC'$ or others are right. Eliminate.
4. Analyze Option 4: By the definition of reflection, the line of reflection ($\overleftrightarrow{EG}$) is perpendicular to the segment joining a point and its image (so $\overleftrightarrow{EG} \perp \overline{CC'}$). Thus, in $\triangle BGC$, $\angle BGC$ is right (since $\overleftrightarrow{EG} \perp \overline{CC'}$), so $\triangle BGC$ is right.

Answer:

D. $\triangle BGC$ is right because $\overleftrightarrow{EG} \perp \overline{CC'}$.