Which of these transformations are isometries? The diagrams are not drawn to scale.
(I) parallelogram EFGH → parallelogram XWVU
The figure shows parallelograms E F G H and X W V U. Segment E F equals 5. Segment F G equals 10. angle E measures 43 degrees. Segment X W equals 5. Segment W V equals 10. angle X measures 43 degrees.
(II) hexagon CDEFGH → hexagon TUVWXY
The figure shows hexagon C D E F G H and hexagon T U V W X Y.
Segment C D equals 6. Segment D E equals 8. Segment E F equals 18. Segment F G equals 8. Segment G H equals 10. Segment H C equals 18. Angle D measures 130 degrees. Angle G measure 102 degrees.
Segment T U equals 3. Segment U V equals 4. Segment V W equals 9. Segment W X equals 4. Segment X Y equals 5. Segment Y T equals 9. Angle U measures 130 degrees. Angle X measure 102 degrees.
(III) triangle EFG → triangle VWU
Two triangles are shown.• Triangle upper E upper F upper G is shown with vertex upper F at the top. Side upper E upper F is labeled 16. Side upper F upper G is labeled 19. Side upper G upper E is labeled 18. Angle upper F is labeled 50 degree sign.
• Triangle upper W upper V upper U is shown with vertex upper W at the bottom. Side upper V upper W is labeled 16. Side upper U upper W is labeled 19. Side upper V upper U is labeled 18. Angle upper W is labeled 50 degree sign.
(1 point)
Responses
I only
I only
II and III only
II and III only
I and III only
I and III only
I, II, and III
I, II, and III
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The correct answer is:
I and III only
The transformation (I) is an isometry because it preserves congruence. The corresponding sides and angles of parallelogram EFGH and parallelogram XWVU are equal, which means that the transformation is an isometry.
The transformation (II) is not an isometry because the corresponding sides and angles of hexagon CDEFGH and hexagon TUVWXY are not equal. The transformation changes the lengths of the sides and the measures of the angles, so it is not an isometry.
The transformation (III) is an isometry because the corresponding sides and angles of triangle EFG and triangle VWU are equal. The transformation preserves congruence, so it is an isometry.
Therefore, the correct answer is (I) only and (III) only.