Question

There is a rectangle ABCD, with sides AB = CD = 32, and sides and BC = DA = 24. The rectangle is rotated 90° clockwise about C, then rotated 90° clockwise about the new location of point that D after the first rotation. What is the length of the path travelled by point A?

Answer 1

Answer

given,

side of rectangle, AB = CD = 32

                              BC = DA = 24

rectangle is rotated 90° clockwise about C.

then rotated 90° clockwise about D.

Path traveled by the point A for first rotation will be in circle with radius AC.

   $D_1 = \dfrac{\theta_1}{360^0}\times \pi \times {AC}^2$

      $AC = \sqrt{24^2+32^2}$

             AC = 40

             θ₁ = 90°

   $D_1 = \dfrac{90^0}{360^0}\times \pi \times {40}^2$

         D₁ = 1256.64

For the second rotation Point A will move in circular path with radius of AD

   $D_2 = \dfrac{\theta_2}{360^0}\pi {AD}^2$

             AD = 24

             θ₁ = 90°

   $D_2 = \dfrac{90^0}{360^0}\times \pi \times {24}^2$

         D₂ = 452.39

total path traveled by the point A

   D = D₁ + D₂

   D = 1256.64 + 452.39

  D = 1709.03