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A regular octagon ABCDEFGH rotates 360 degrees clockwise about its center.

After how many 45-degree increments will A' (the image of A) coincide with F?

Answer :

Answer:

If the octagon is clockwise, then after 5 times 45 increments A' will coincide with F.

If the octagon is counterclockwise, then after 3 times 45 increments A' will coincide with F.

Step-by-step explanation:

Case 1: Let the octagon is clockwise

It is given that a regular octagon ABCDEFGH rotates 360 degree clockwise about it's center.

There are 8 vertices and the central angle between two consecutive vertices is 45 degrees.

Central angle between AB is 45 degree.

Central angle between AC is

[tex]\text{Angle between AB}+\text{Angle between BC}=45^{\circ}+45^{\circ}=2\times 45^{\circ}[/tex]

Central angle between AD is

[tex]\text{Angle between AB}+\text{Angle between BC}+\text{Angle between CD}=45^{\circ}+45^{\circ}+45^{\circ}=3\times 45^{\circ}[/tex]

Similarly,

Central angle between AE is

[tex]4\times 45^{\circ}[/tex]

Central angle between FA is

[tex]5\times 45^{\circ}[/tex]

Therefore after 5 times 45 increments A' will coincide with F.

Case 2: Let the octagon is counterclockwise

[tex]FA=AH+HG+GF[/tex]

[tex]FA=45^{\circ}+45^{\circ}+45^{\circ}[/tex]

[tex]FA=135^{\circ}[/tex]

[tex]FA=3\times 45^{\circ}[/tex]

Therefore after 3 times 45 increments A' will coincide with F.

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Rewritten by : Barada

Going from A to B means you've rotated 45 degrees. Then from B to C, you've gone 45 more degrees. So how many times do you have to do this to land on F?

B - 1×45 degrees
C - 2×45 degrees
D - 3×45 degrees
E - 4×45 degrees
F - 5×45 degrees

5×45 = 225 degrees