The perimeter of rectangle ABCD is 30 cm. The bisectors of corners A and D intersect at point M adjacent to side BC.

The perimeter of rectangle ABCD is 30 cm. The bisectors of corners A and D intersect at point M adjacent to side BC. Find the sides of the rectangle.

The bisectors of right angles divide the angle into two angles of 45 °. Since the angles C and B are straight, then by the property of the sum of acute angles of a right-angled triangle, the angles DMC and AMB are respectively equal to 45 °, therefore, the triangles CDM and MAB are isosceles and DC = CM = MB = AB. So CB = 2DC.
Let DC = x, then CB = 2x.
P = 2 (DC + CB) = 2 (x + 2x) = 6x.
But because the perimeter is 30cm, then
30 = 6x
X = 30: 6
x = 5cm
2x = 5 × 2 = 10cm
Answer: the sides are 5cm and 10cm