The square with vertices (-a, -a), (a, -a), (-a, a), (a, a) is cut by the line y = x/3 into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by a equals what? Express your answer in simplified radical form.
+128474 Call the length of the top side of one of the quadrilaterals, 2a
The length of the right side of this quadilateral (a - a/3) = (2/3)a = (2a)/3
The length of the bottom of the quadrilateral = sqrt [ ( a - - a)^2 + (a/3 - - a/3)^2 ] =
sqrt [ 4a^2 + (4/9)a^2 ] = a sqrt [ 36 + 4]/3 = (a/3)sqrt (40)
The left side of the quadrilateral = (a - - a/3) = (4a)/3
Perimeter of quadrilateral =
a [ 2 + 2/3 + sqrt (40)/3 + 4/3 ] = a [ 4 + sqrt (40)/3 ] = a [ 12 + 2sqrt (10) ] / 3
Dividing this by a = [ 12 + 2sqrt (10)] / 3 = 4 + (2/3)sqrt (10)
