ABCD-rectangle. M-midpoint of side BC. It is known that lines MA and MD are mutually perpendicular.
September 22, 2021 | education
| 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;
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