The square with vertices (– a, – a), (a, – a), (– a, a) and (a, a) is cut by the line y = x/2 into congruent quadrilaterals. What is the number of units in the perimeter of each quadrilateral?
Since there are two congruent quadrilaterals, each quadrilateral will get one-half of the perimeter of the square. The perimeter of the square is 8a, so each quadrilateral will have 4a.
Each quadrilateral also gets the length of the diagonal cut.
The left-hand end of the diagonal will occur when x = -a.
When x = -a, y = -a/2, so the point is (-a, -a/2).
The right-hand end of the diagonal occurs when x = a.
When x = a, y = a/2, so the point is (a, a/2).
Using the distance formula to find the length from (-a, -a/2) to (a, a/2):
distance = sqrt( (a - a)2 + (a/2 - -a/2)2 ) = sqrt( (2a)2 + (a)2 ) = sqrt( 5a2 ) = sqrt(5)·a.
So, the perimeter of each quadrilater is 4a + sqrt(5)·a or ( 4 + sqrt(5) )·a