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1)Bprime is the image of B rotated 60 degrees counter-clockwise about A. Given AB = 6sqrt3 find BBprime.

 

2)Through how many different positive angles less than 360 degrees is it possible to rotate a regular icosagon (20-gon) clockwise about its center such that its image coincides with the original icosagon?

 

3)A translation maps A to Aprime and B to Bprime. We know  AAprime = 3, AB = 4 and ABprime = 5.  Find BAprime.

 

4)A rectangle that is not a square is rotated counterclockwise about its center. What is the minimum positive number of degrees it must be rotated until it coincides with its original figure?

 

5)A regular pentagon is rotated counterclockwise about its center by a nonzero angle. What is the minimum number of degrees it must be rotated until it coincides with its original position?

Picture:https://latex.artofproblemsolving.com/e/6/6/e66e869ca42cd75d48a55b84c9acc3815e694262.png

 

6)Jessica wants to add one more dot to the diagram below so that the four dots form the four vertices of a parallelogram. In how many possible locations can she put her dot?

Picture:https://latex.artofproblemsolving.com/e/a/5/ea5e9a9dac5003a72e6e808494f935c99bdbf56a.png

 

7)A circle with diameter 2 is translated 5 units. What is the perimeter of the region swept out by the circle?

Picture:https://latex.artofproblemsolving.com/1/4/9/14901c49f11e8cb2655024bbe5377b0eab31f100.png

 

8)A pentagon ABCDE is translated north by 40 units to pentagon A1B1C1D1E1. Pentagon A1B1C1D1E1 is then translated east by 50 units to pentagon A2B2C2D2E2. We know the perimeter of pentagon ABCDE is 30 units. Find CC2.

 

9)A and B are two points on a plane. A translation maps A to B and B to Bprime. Another translation maps B to A  and A to Aprime. We know AprimeBprime = 16. Find AB.

 

10)OPQRSTUVWX is a regular decagon (10-gon). A rotation of theta degrees about U maps X to R. Given theta < 180 degrees, find theta.

Picture:https://latex.artofproblemsolving.com/3/e/c/3ec0f4ea12e566e6e5007d09b0f3ddc79e998401.png

 

11)Let ABCD be a square of side length 1. Let P be a point on side CD such that angle DAP = 20 degrees. Let Q be a point on side BC such that angle BAQ = 25 degrees. Find the perimeter of triangle CPQ.

Picture:https://latex.artofproblemsolving.com/d/d/9/dd9e479748af02fbab6e119ffde36405aa73224b.png

FiestyGeco  Apr 22, 2018
 #1
avatar+88871 
+1

1)B' is the image of B rotated 60 degrees counter-clockwise about A. Given AB = 6sqrt3 find BB'.

 

Let A  lie at  (0,0)

Let B lie at  (6√3 , 0)

 

Then  B'  will  be   (6√3 cos (60), 6√3 sin (60) )  =  (3√3, 9) 

 

So....BB'  is  √ [ ( 6√3 - 3√3)^2  + 9^2 ]   =  √ [ (3√3)^2  + 9^2 ]  = √ [ (√27)^2 + 81 ] =

 

√ [ 27 + 81 ]  =  √108  =  6√3

 

 

cool cool cool

CPhill  Apr 22, 2018
 #2
avatar+88871 
+1

4)A rectangle that is not a square is rotated counterclockwise about its center. What is the minimum positive number of degrees it must be rotated until it coincides with its original figure?

 

180°

 

 

cool cool cool

CPhill  Apr 22, 2018
 #3
avatar+88871 
+1

7)A circle with diameter 2 is translated 5 units. What is the perimeter of the region swept out by the circle?

 

I'm assuming  that the "swept out "  region  is the one between both circles

 

If so.....its  perimeter is  composed of the perimeter of a  circle with a radius of 1  plus  two sides of 5 units each

 

So we have   

 

2pi (1)  + 2* 10   =

 

[ 2pi + 10 ]  units

 

 

 

cool cool cool

CPhill  Apr 22, 2018

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