In the rectangle ABCD, the bisectors of angles A and D are drawn, which intersect at point M

In the rectangle ABCD, the bisectors of angles A and D are drawn, which intersect at point M, which lies on side B. Find the perimeter ABCD if AB = 6cm.

Consider a rectangle ABCD with bisectors AM and DM of angles A and D.

Since the angle is BAC = 90 ° AНD AM is the bisector of this angle, then BAM = MAD = 45 °. Hence, BAM = BMA = 45 °.

Since the angle CDA = 90 ° AНD DM is the bisector of this angle, then CDM = MDA = 45 °. Hence, DMC = CDM = 45 °.

Therefore, triangles ABM and MCD are right-angled and isosceles. It means that:

AB = BM = 6,

CM = CD = AB = 6.

Therefore, the perimeter P of rectangle ABCD is:

P = 2 * (AB + BC) = 2 * (AB + BM + CM) = 2 * (6 + 6 + 6) = 36.

Answer: P = 36 cm.