Question

rotating δpqr by 180° clockwise about the origin, we get its image δpqr as shown below. note that δpqr has vertices p(-5, 7), q(-6, 3), and r(-8, 1). also, note that δpqr has vertices p(5, -7), q(6, -3), and r(8, -1). complete the following. scratch area (not part of answer) (a) suppose each pair of segments below have the same length. find each length. give exact answers (not decimal approximations). cp = cp = units cq = cq = units cr = cr = units (b) suppose the angles below have the same measure. choose the correct angle measure. use the protractor provided, as necessary. m∠pcp = m∠qcq = m∠rcr = select (c) choose the correct pair of statements about the rotation. not every point on the original figure is the same distance from the center of rotation as its image. not all angles formed by a point and its image, with the vertex at the center of rotation, are congruent. each point on the original figure is the same distance from the center of rotation as its image. not all angles formed by a point and its image, with the vertex at the center of rotation, are congruent. not every point on the original figure is the same distance from the center of rotation as its image. all angles formed by a point and its image, with the vertex at the center of rotation, are congruent. each point on the original figure is the same distance from the center of rotation as its image. all angles formed by a point and its image, with the vertex at the center of rotation, are congruent.

Explanation:

Step1: Calculate $CP$ (distance formula)

$CP = \sqrt{(-5-0)^2 + (7-0)^2} = \sqrt{25 + 49} = \sqrt{74}$

Step2: Calculate $CQ$ (distance formula)

$CQ = \sqrt{(-6-0)^2 + (3-0)^2} = \sqrt{36 + 9} = \sqrt{45}$

Step3: Calculate $CR$ (distance formula)

$CR = \sqrt{(-8-0)^2 + (1-0)^2} = \sqrt{64 + 1} = \sqrt{65}$

Step4: Identify rotation angle

Rotation is $180^\circ$, so angles are $180^\circ$

Step5: Select rotation properties

Rotations preserve distance and rotation angles, so the last option is correct.

Answer:

(a) $CP = CP' = \sqrt{74}$ units
$CQ = CQ' = \sqrt{45}$ units
$CR = CR' = \sqrt{65}$ units
(b) $180^\circ$
(c) Each point on the original figure is the same distance from the center of rotation as its image. All angles formed by a point and its image, with the vertex at the center of rotation, are congruent.