Question

look at this diagram:
diagram of triangle ghi with point j on gh
which diagram shows δghi rotated 90° counterclockwise about j?
two diagrams below, one with purple triangle ghi and one with other rotated triangle

Explanation:

Step1: Recall Rotation Rules

A 90° counterclockwise rotation about a point \( J \) means that for each point \( (x,y) \) relative to \( J \), the new coordinates \( (x',y') \) are given by \( (-y + x_J, x + y_J) \) (if \( J \) is at \( (x_J,y_J) \)). Alternatively, visually, the segment from \( J \) to a vertex should be perpendicular (90°) and same length after rotation, with counterclockwise direction (left - turn).

Step2: Analyze the First Diagram

In the first diagram (the upper one with \( G', H', I' \)):

• Check the rotation of \( H \) about \( J \). The segment \( JH \) and \( JH' \) should be perpendicular (90°) and same length. Visually, the angle between \( JH \) and \( JH' \) appears 90° counterclockwise. Similarly for \( JG \) and \( JG' \), and \( JI \) and \( JI' \). The triangle \( \Delta G'H'I' \) is a 90° counterclockwise rotation of \( \Delta GHI \) about \( J \) as the sides from \( J \) to each vertex are rotated 90° counterclockwise and lengths are preserved.

Step3: Analyze the Second Diagram

In the second diagram, the rotation direction or the perpendicularity (90° angle) between original and rotated segments from \( J \) does not match a 90° counterclockwise rotation. The positions of \( G', H', I' \) relative to \( J \) do not show the correct 90° counterclockwise turn.

Answer:

The upper diagram (the first one with \( G', H', I' \) in purple) shows \( \Delta GHI \) rotated \( 90^\circ \) counterclockwise about \( J \).