Question

the transformation we saw in both exercises #1 and #2 are known as rotations. rotations of a point are def by a center point, a circle, and an angle whose vertex is at the center of the circle.
rotations
the rotation of point ( a ) about (or around) point ( c ) in a specific direction (c.w. or c.c.w.) by a specific angle is accomplished by drawing a circle with center at ( c ) and radius of ( overline{ca} ). the image point is located on the circle such that ( mangle aca ) is equal to the specified angle in the specified direction.
exercise #3: circle ( o ) shown below has points ( e,f,g ), and ( h ) marked. rays are drawn from ( o ) through each point and the angles created by adjacent rays are shown. answer the following:
(a) what two rotations about point ( o ) will map point ( e ) onto point ( g )?
(b) what two rotations about point ( o ) will map point ( g ) onto point ( h )?
(c) using a straightedge only, locate the image of point ( e ) after a ( 180^{circ} ) rotation about point ( o ). label it point ( i ). why does the direction not need to be specified?
exercise #4: in the diagram below, point ( n ) is the image of point ( m ) after a clockwise rotation about point ( q ) by ( 125^{circ} ). point ( p ) is the image of point ( n ) after a clockwise rotation about point ( q ) by ( 55^{circ} ).
(a) label the two angles given in the problem on the diagram.
(b) what three line segments must have the same length (be congruent) based on the information in the problem. explain.
(c) if point ( m ) was mapped to point ( p ) using a single clockwise rotation about point ( q ), through what angle should the rotation take place?
(d) what does the answer to (c) tell you about the points ( m,q ), and ( p )?

Explanation:

Step1: Calculate the rotation angle from \(E\) to \(G\)

Clockwise rotation: \(95^{\circ}+54^{\circ}=149^{\circ}\)
Counter - clockwise rotation: \(360^{\circ}-149^{\circ}=211^{\circ}\)

Step2: Calculate the rotation angle from \(G\) to \(H\)

Clockwise rotation: \(90^{\circ}\)
Counter - clockwise rotation: \(360^{\circ}-90^{\circ}-54^{\circ}=216^{\circ}\)

Step3: Locate \(I\) for \(180^{\circ}\) rotation of \(E\)

A \(180^{\circ}\) rotation is a straight line. Since \(180^{\circ}\) rotation results in the same position regardless of direction (clockwise or counter - clockwise as it's a half - turn), draw a straight line from \(E\) through \(O\) and mark \(I\) on the opposite side.

Step4: Label angles for Exercise #4(a)

Label \(\angle MQN = 125^{\circ}\) (rotation from \(M\) to \(N\)) and \(\angle NQP=55^{\circ}\) (rotation from \(N\) to \(P\))

Step5: Find congruent segments for Exercise #4(b)

By rotation properties (\(MQ = NQ\) as \(N\) is image of \(M\) rotated about \(Q\); \(NQ = PQ\) as \(P\) is image of \(N\) rotated about \(Q\)), so \(MQ=NQ = PQ\)

Step6: Calculate rotation angle for Exercise #4(c)

Total clockwise rotation from \(M\) to \(P\) is \(125^{\circ}+55^{\circ}=180^{\circ}\)

Step7: Analyze points \(M\), \(Q\), \(P\) for Exercise #4(d)

Since \(MQ = PQ\) and rotation is \(180^{\circ}\), \(M\), \(Q\), \(P\) are collinear with \(Q\) as the mid - point.

Answer:

(a) \(149^{\circ}\) clockwise and \(211^{\circ}\) counter - clockwise rotations map \(E\) to \(G\)
(b) \(90^{\circ}\) clockwise and \(216^{\circ}\) counter - clockwise rotations map \(G\) to \(H\)
(c) Locate \(I\) by drawing a straight line from \(E\) through \(O\); direction not needed as \(180^{\circ}\) rotation (half - turn) is same in either direction
(a) Label \(\angle MQN = 125^{\circ}\) and \(\angle NQP = 55^{\circ}\)
(b) \(MQ\), \(NQ\), \(PQ\) (rotation preserves length)
(c) \(180^{\circ}\)
(d) \(M\), \(Q\), \(P\) are collinear with \(Q\) as the mid - point.