The square with vertices (a,a), (-a,a), (-a,-a), (a,-a) is cut by the line y=x/2 into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by a equals what? Express your answer in simplified radical form.
+23246 If you draw the square with vertices: (a, a) (-a, a) (a, -a) (-a, a)
and the line y = x/2 this line will pass through the points (a, a/2) and (-a, -a/2).
Consider the quadrilateral whose enpoints are (a, a/2) (a, a) (-a, a) (-a, -a/2).
The distance from (a, a/2) to (a, a) is a/2.
The distance from (a, a) to (-a, a) is 2a.
The distance from (-a, a) to (-a, -a/2) is 3/2·a
To find the distance from (-a, -a/2) to (a, a/2) we can use the distance formula:
distance = sqrt[ (a - a)2 + (a/2 - -a/2)2 ] = sqrt[ (2a)2 + (a)2 ] = sqrt[ 4a2 + a2 ] = sqrt( 5a2 ) = a·sqrt(5).
Adding these togerther: a/2 + 2a + 3/2·a + a·sqrt(5) = 4a + a·sqrt(5)
Dividing this by a gives 4 + sqrt(5).
