Question

the image of △abc after a reflection across ec is △abc. which triangle must be a right triangle and why? △abc is right because it is the image of △abc. △adc is right because aa intersects ac at a. △bcc is right because b lies of the line of reflection. △bgc is right because ec ⊥ cc.

Explanation:

Step1: Recall reflection property

In a reflection, the line of reflection is the perpendicular - bisector of the segments connecting pre - image and image points.

Step2: Analyze each option

For $\triangle A'B'C'$, just being an image doesn't guarantee it's a right - triangle. For $\triangle ADC$, the intersection of $AA'$ and $AC$ at $A$ doesn't make it a right - triangle. For $\triangle BCC'$, point $B$ lying on the line of reflection doesn't make it a right - triangle. For $\triangle BGC$, since the line of reflection $\overline{EG}$ is perpendicular to the segment $\overline{CC'}$ (by the property of reflection, the line of reflection is perpendicular to the segment joining a point and its image), $\angle BGC = 90^{\circ}$ and $\triangle BGC$ is a right - triangle.

Answer:

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