Question

rotating $\triangle lmn$ by $90^circ$ clockwise about the origin, we get its image $\triangle lmn$ as shown below.
note that $\triangle lmn$ has vertices $l(1, 8)$, $m(3, 5)$, and $n(6, 6)$.
also, note that $\triangle lmn$ has vertices $l(8, -1)$, $m(5, -3)$, and $n(6, -6)$.
complete the following.
scratch area (not part of answer)
(a) suppose the angles below have the same measure. choose the correct angle
measure. use the protractor provided, as necessary.
$mangle lcl = mangle mcm = mangle ncn = $ select
(b) suppose each pair of segments below have the same length.
find each length. give exact answers (not decimal approximations).
$cl = cl = \square$ units
$cm = cm = \square$ units
$cn = cn = \square$ units
(c) choose the correct pair of statements about the rotation.
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.
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.

Explanation:

Step1: Identify rotation angle

The problem states the rotation is $90^\circ$ clockwise about the origin, so the angle between each point and its image (with vertex at origin) is $90^\circ$.

Step2: Calculate $CL$ (distance from origin to $L$)

Use distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, origin $(0,0)$, $L(1,8)$:
$\sqrt{(1-0)^2+(8-0)^2}=\sqrt{1+64}=\sqrt{65}$

Step3: Calculate $CM$ (distance from origin to $M$)

Origin $(0,0)$, $M(3,5)$:
$\sqrt{(3-0)^2+(5-0)^2}=\sqrt{9+25}=\sqrt{34}$

Step4: Calculate $CN$ (distance from origin to $N$)

Origin $(0,0)$, $N(6,6)$:
$\sqrt{(6-0)^2+(6-0)^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}$

Step5: Verify rotation properties

Rotations preserve distances (each point and its image are same distance from origin) and the angle of rotation is consistent for all points, so all such angles are congruent.

Answer:

(a) $90^\circ$
(b)
$CL = CL' = \sqrt{65}$ units
$CM = CM' = \sqrt{34}$ units
$CN = CN' = 6\sqrt{2}$ units
(c) 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.