it is about a square and area

Question

Points E and F are chosen on sides BC and AB respectively of the square ABCD
so that the lines DE and DF divide the square into three regions of equal area. The
sides of the square have length 1,then DF=√a.BF. Find the value of a.

Answer 1

Answer removed by provider

Answer 2

The three areas, that of triangles ADF and DEC and that of quadrilateral DEBF are equal. The two triangles have a square edge as a side and AF or CE as the side perpendicular to the square edge. Call their areas S. Then S = 1/2 AF = 1/2 CE, thus, AF=CE. The area of the quadrilateral DEBF = area of the square - the area of the two triangles = 1 - 2S. Since the three areas are equal, S = 1-2S or S = 1/3. Therefore AF = CE = 2/3. BF = 1/3 and DF = √(1+AF²) = √13 / 3 = BF √13. Thus a = √13.