Question

1. lines k and m are perpendicular and intersect at point r. points q and q both lie on line k such that qr = rq. which statement must be true? a) a reflection over line k maps point q onto q b) a reflection over line m maps point q onto q c) a translation of length qr maps point q onto q d) a translation of length ½qr maps point q onto q 6. which transformations will always produce congruent figures? a) dilation c) rigid motions b) transformation d) non - rigid motions 7. specify a sequence of transformations that will map abcd to pqrs. a) reflection and rotation b) rotation and translation c) reflection and translation d) rotation and dilation 8. congruent quadrilaterals jklm and pqrs are shown. which sequence of transformations maps jklm onto pqrs? a) a reflection over the x - axis followed by a translation 2 units to the right b) a reflection over the y - axis followed by a translation 8 units down and 5 units left c) a rotation 180° clockwise about the origin followed by a reflection over the y - axis d) a rotation 90° clockwise about point m followed by a translation 8 units down and 3 units left 9. points m and m are two distinct points that lie on a circle a. which statement must be true? a) point m can be reflected over am to map onto point m. b) point m can be rotated about point a to map onto point m. c) point m can be translated to a distance of am to map onto point m. d) point m can be dilated by a scale factor of am centered at point a to map onto point m.

Explanation:

Step1: Recall reflection property

If two points Q and Q' are equidistant from a line k (QR = RQ') and lie on a line perpendicular to k, a reflection over line k maps Q onto Q'.

Step2: Analyze option B

Line m is perpendicular to k, so reflection over m won't map Q onto Q'.

Step3: Analyze option C

Translation won't map Q onto Q' as translation moves points in a straight - line direction, not based on the perpendicular bisector property.

Step4: Analyze option D

Same as option C, translation of length QR or ½QR won't map Q onto Q' based on the given perpendicular and equidistant condition.

Step1: Recall transformation types

Dilation changes the size of a figure, so it doesn't always produce congruent figures.

Step2: Understand rigid motions

Rigid motions (translations, rotations, reflections) preserve distance and angle measures, always producing congruent figures.

Step3: Analyze option B

The term "transformation" is too general and includes non - rigid motions like dilation.

Step4: Analyze option D

Non - rigid motions change the size or shape of a figure and don't produce congruent figures.

Step1: Observe the orientation and position

The orientation of ABCD and PQRS is different. A rotation is needed to change the orientation.

Step2: Observe the position change

After rotation, a translation is needed to move the figure to the correct position.

Step3: Analyze option A

Reflection and rotation may not be sufficient to map ABCD to PQRS as the position may not be correct after just these two.

Step4: Analyze option C

Reflection and translation may not account for the orientation change.

Step5: Analyze option D

Dilation changes the size, and the figures are congruent, so dilation is not needed.

Step1: Analyze option A

Reflecting over the x - axis and then translating 2 units to the right won't map JKLM onto PQRS.

Step2: Analyze option B

Reflecting over the y - axis and then translating 8 units down and 5 units left won't map JKLM onto PQRS.

Step3: Analyze option C

Rotating 180° clockwise about the origin and then reflecting over the y - axis won't map JKLM onto PQRS.

Step4: Analyze option D

By rotating 90° clockwise about point M and then translating 8 units down and 3 units left, JKLM can be mapped onto PQRS.

Answer:

A. A reflection over line k maps point Q onto Q'