ABCD-rectangle. M-midpoint of side BC. It is known that lines MA and MD are mutually perpendicular.

ABCD-rectangle. M-midpoint of side BC. It is known that lines MA and MD are mutually perpendicular. And that the perimeter (Равсd) of the rectangle ABCD = 24. Find his sides

ΔABM = ΔMCD are equal, because <B = <C = 90 °, BA = CD are equal as opposite sides of the rectangle, and BM = MD (point M divides BC in half).

<BMC = <BMA + <AMD + <DMC (1);

180 ° = <BMA + 90 ° + <DMC (2);

180 ° = <BMA + 90 ° + <BMA;

90 ° = 2 * <BMA;

<BMA = 45 °; <DMC = 45 °;

<BAM = 180 ° – <ABM – <BMA = 180 ° – 90 ° – 45 ° = 45 °.

ΔABM – isosceles, isosceles and ΔMCD:

AB = BM; DC = MC;

BC = BM + MC = AB + DC = 2 * AB;

AD = BC = 2 * AB

PABCD = AB + BC + CD + AD = AB + 2 * AB + AB + 2 * AB = 6 * AB.

24 = 6 * AB;

Answer: AB = CD = 4; BC = AD = 4 * 2 = 8;