Question

the image of ( \triangle abc ) after a reflection across ( overleftrightarrow{ec} ) is ( \triangle abc ). which triangle must be a right triangle and why? ( \triangle abc ) is right because it is the image of ( \triangle abc ). ( \triangle adc ) is right because ( overline{aa} ) intersects ( overline{ac} ) at ( a ). ( \triangle bcc ) is right because ( b ) lies of the line of reflection. ( \triangle bgc ) is right because ( overleftrightarrow{ec} perp overline{cc} ).

Explanation:

Step1: Analyze the property of reflection

When a figure is reflected across a line, the line of reflection is the perpendicular bisector of the segment joining a point and its image.

Step2: Check each option

• For $\triangle A'B'C'$: Just being an image does not imply it is right - angled.
• For $\triangle ADC$: Intersection of $AA'$ and $AC$ at $A$ does not make it right - angled.
• For $\triangle BCC'$: Since $B$ lies on the line of reflection $EG$, and $EG$ is the perpendicular bisector of $CC'$ (by reflection property), so $\angle BGC = 90^{\circ}$ (as $EG\perp CC'$).

Answer:

$\triangle BGC$ is right because $EG\perp CC'$.