Question

draw the image of △abc after a reflection across line ℓ.

Explanation:

To determine the correct reflection of \(\triangle ABC\) across line \(\ell\) (which appears to be the line \(y = x\) or a line with a slope of \(1\) based on the grid), we use the property of reflections: the distance from each vertex to the line of reflection should be equal to the distance from its image to the line of reflection, and the line of reflection is the perpendicular bisector of the segment joining a vertex and its image.

Step 1: Recall the reflection property

For a reflection across a line with slope \(1\) (like \(y = x\)), the coordinates \((x, y)\) of a point are transformed to \((y, x)\). Alternatively, geometrically, we can count the horizontal and vertical distances from each vertex to the line \(\ell\) and plot the image such that these distances are mirrored on the other side of \(\ell\).

Step 2: Analyze each option

• First option (red triangle \(A'B'C'\)): The distances from \(A, B, C\) to \(\ell\) do not match the distances from \(A', B', C'\) to \(\ell\). The shape and orientation are inconsistent with a reflection over \(\ell\).
• Second option (red \(A', B', C'\)): The segment \(A'B'\) is horizontal, but the reflection of a non - horizontal segment over a line with slope \(1\) should not be horizontal. The position of \(C'\) is also incorrect.
• Third option (red \(A', B', C'\)): The distances from \(A\) to \(\ell\) and \(A'\) to \(\ell\), \(B\) to \(\ell\) and \(B'\) to \(\ell\), \(C\) to \(\ell\) and \(C'\) to \(\ell\) are equal. The orientation of the triangle is consistent with a reflection over \(\ell\) (the line of reflection is the perpendicular bisector of \(AA'\), \(BB'\), and \(CC'\)).
• Fourth option (red triangle): The distances and the shape of the triangle do not match the expected reflection over \(\ell\).

Answer:

The third option (the one with red \(A'\), \(B'\), \(C'\) where \(A'\) and \(B'\) are horizontally aligned below the line \(\ell\) and \(C'\) is below \(C\) with the correct distance to \(\ell\)) is the correct reflection. (If we assume the options are labeled from top to bottom as Option 1, Option 2, Option 3, Option 4, Option 5, then the third option (the middle - looking one among the red - triangle options) is correct. Based on the visual, the third option (the one with \(A'\), \(B'\) horizontal and \(C'\) below \(C\) with the correct reflection - based positioning) is the answer. If we need to label it as per the options (assuming the first red - triangle option is Option 1, the next Option 2, etc.), the correct one is the third option (the one with \(A'\), \(B'\) and \(C'\) such that the reflection property holds).