Question

rotating δlmn by 90° clockwise about the origin, we get its image δlmn as shown below. note that δlmn has vertices l(-3, -8), m(-6, -6), and n(-7, -2). also, note that δlmn has vertices l(-8, 3), m(-6, 6), and n(-2, 7). 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. 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. 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.

Explanation:

Part (a)

Step1: Recall rotation angle

A \( 90^\circ \) clockwise rotation about the origin means the angle between a point, the center of rotation (origin), and its image is \( 90^\circ \). So \( m\angle LCL' = m\angle MCM' = m\angle NCN' = 90^\circ \).

To find the length of \( CL \), \( CM \), and \( CN \), we use the distance formula from a point \((x,y)\) to the origin \((0,0)\), which is \( d = \sqrt{(x - 0)^2+(y - 0)^2}=\sqrt{x^2 + y^2} \).

Step1: Calculate \( CL \) (for \( L(-3,-8) \))

\( CL=\sqrt{(-3)^2+(-8)^2}=\sqrt{9 + 64}=\sqrt{73} \)

Step2: Calculate \( CM \) (for \( M(-6,-6) \))

\( CM=\sqrt{(-6)^2+(-6)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2} \)

Step3: Calculate \( CN \) (for \( N(-7,-2) \))

\( CN=\sqrt{(-7)^2+(-2)^2}=\sqrt{49 + 4}=\sqrt{53} \)

• In a rotation, the angle between a point, the center of rotation, and its image (rotation angle) is constant (congruent for all such angles) because it's a rigid transformation with a fixed rotation angle.
• Also, a rotation is a rigid transformation, so the distance from each point to the center of rotation is preserved (each point and its image are equidistant from the center).

So the correct pair is: "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."

Answer:

\( 90^\circ \)

Part (b)