Question

1. determine whether each transformation preserves only length, angle measure, or both. circle all that apply. a. rotation preserves: -length- -angle measure- -both- b. dilation ( k = 1 ) preserves: -length- -angle measure- -both- c. reflection preserves: -length- -angle measure- -both- d. translation then a dilation with ( k = 2 ) preserves: -length- -angle measure- -both- 7. which transformation preserves both distance and angle measure? circle one. a. ( (x,y)\to(-y + 4,x - 6) ) b. ( (x,y)\to(-2y + 4,x - 6) ) c. ( (x,y)\to(2x - 4,2y - 6) ) d. ( (x,y)\to(2x - 4,y - 6) )

Explanation:

Step1: Analyze Rotation

Rotation is a rigid transformation. Rigid transformations preserve both length and angle measure.

Step2: Analyze Dilation \(k = 1\)

A dilation with \(k = 1\) is the identity transformation. It does not change the figure, so it preserves both length and angle measure.

Step3: Analyze Reflection

Reflection is a rigid transformation. Rigid transformations preserve both length and angle measure.

Step4: Analyze Translation then Dilation \(k = 2\)

Translation is a rigid transformation (preserves length and angle measure), but dilation with \(k=2\) changes the length (scales it by 2). However, dilation preserves angle measure.

For problem 7:

Step1: Analyze Option A

The transformation \((x,y)\to(-y + 4,x-6)\) is a rotation (by \(90^{\circ}\) counter - clockwise) followed by a translation. Rotation and translation are rigid transformations. Rigid transformations preserve both distance (length) and angle measure.

Step2: Analyze Option B

The transformation \((x,y)\to(-2y + 4,x-6)\) is a non - rigid transformation (because of the factor of 2 multiplying \(y\)). It does not preserve distance.

Step3: Analyze Option C

The transformation \((x,y)\to(2x-4,2y - 6)\) is a dilation (\(k = 2\)) followed by a translation. Dilation with \(k
eq1\) changes the distance.

Step4: Analyze Option D

The transformation \((x,y)\to(2x-4,y - 6)\) is a non - rigid transformation (because of the factor of 2 multiplying \(x\)). It does not preserve distance.

Answer:

a. BOTH
b. BOTH
c. BOTH
d. ANGLE MEASURE

1. A. \((x,y)\to(-y + 4,x-6)\)