Question

1. the quadrilateral math was rotated 90 degrees counterclockwise followed by a translation to produce the quadrilateral math.

Explanation:

Step1: Recall Rotation and Translation Properties

Rotations and translations are rigid transformations, so the length of each line segment remains unchanged. This means the length of a segment in MATH is equal to the length of the corresponding segment in M'A'T'H'.

Step2: Match Segments by Corresponding Parts

• Line segment MA in MATH corresponds to line segment M'A' in M'A'T'H'. Since rigid transformations preserve length, MA should match the segment labeled 'a' (as the red quadrilateral's side 'a' corresponds to MA after transformation? Wait, actually, looking at the grid, we need to check the lengths. Wait, the key is that rotation and translation don't change side lengths. So each segment in MATH has the same length as its image in M'A'T'H'. So we need to match the segments: MA corresponds to M'A', AT to A'T', TH to T'H', HM to H'M'. Now, looking at the red quadrilateral (MATH before transformation?) Wait, no, the green is MATH, red is M'A'T'H'? Wait, the problem says MATH was rotated 90 CCW then translated to get M'A'T'H'. So the length of MA (in MATH) should equal the length of M'A' (in M'A'T'H'), which is the segment labeled 'a' in the red quadrilateral. Similarly, AT in MATH (green) should match A'T' (red), which is segment 'b'? Wait, maybe better to calculate the length using distance formula or count grid units. Let's assume the grid is 1 unit per square. For example, if MA in green (MATH) has length equal to the segment 'a' in red (M'A'T'H'), then we match them. Since rigid transformations preserve length, each segment's length is the same as its image. So:
• Line segment MA: matches 'a' (same length)
• Line segment AT: matches 'b' (same length)
• Line segment TH: matches 'c' (same length)
• Line segment HM: matches 'd' (same length)

Wait, maybe the table is to match each segment of MATH (MA, AT, TH, HM) to the corresponding segment in M'A'T'H' (which are labeled a, b, c, d). Since rotation and translation are rigid, lengths are equal, so MA corresponds to the segment with length equal to 'a', AT to 'b', TH to 'c', HM to 'd'? Wait, maybe the red quadrilateral's sides are a, b, c, d, and the green is MATH. So after transformation, MATH becomes M'A'T'H' (red), so MA (green) becomes M'A' (red, segment a), AT (green) becomes A'T' (red, segment b), TH (green) becomes T'H' (red, segment c), HM (green) becomes H'M' (red, segment d). So we fill the table by matching each MATH segment to its image segment (a, b, c, d) since length is preserved.

Answer:

Assuming the table is to match each segment of MATH (MA, AT, TH, HM) to the corresponding segment in M'A'T'H' (a, b, c, d) due to rigid transformations (length preservation):

Line segment MA	a
Line segment AT	b
Line segment TH	c
Line segment HM	d

(Note: The exact matching depends on the grid lengths, but the key is rigid transformations preserve segment lengths, so each segment in MATH has the same length as its image in M'A'T'H', hence matching to the corresponding labeled segment (a, b, c, d) with the same length.)