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.
+2668 The perimeter of the trapezoid is the sum of the sides + the sum of the line cutting the square.
The length of the sides (not including the line) is \(4a\). One of the sides is 2a (labeled a in the diagram), and the other two sides (b and c in the diagram) will always add to 2a.
We can then use the Pythagorean Theorem for the length of the bisecting line, and we find that its length is \(a \sqrt 5 \).
Thus, the perimeter of the shape is \(a(4 + \sqrt 5)\). Can you do the rest?
Diagram:
