Задание 33. Дан прямоугольник ABCD. Заполните пропуски в таблице.

Решение:

1. S_ABCD = AB × AD = 6 × 8 = 48
2. OM = \(\sqrt{OA^2 - AM^2}\) = \(\sqrt{5^2 - 3^2}\) = 4
3. S_ABO = 1/2 × AB × OM = 1/2 × 6 × 5 = 15
4. S_AMC = 1/2 × AM × CD = 1/2 × 3 × 8 = 12
5. AD = S_ABCD / AB = 160 / 10 = 16
6. OM = \(\sqrt{OA^2 - AM^2}\) = \(\sqrt{10^2 - 6^2}\) = 8
7. S_ABO = 1/2 × AB × OM = 1/2 × 10 × 8 = 40
8. S_AMC = 1/2 × AM × CD = 1/2 × 6 × 10 = 30
9. S_ABO = 30, AB = 8, OM = 2S_ABO / AB = 2 × 30 / 8 = 7.5
10. AD = 2 × OM = 2 × 7.5 = 15
11. S_ABCD = AB × AD = 8 × 15 = 120
12. S_AMC = 1/2 × AM × CD = 1/2 × 7.5 × 8 = 30
13. OM = 4, AD = 10, AM = AD / 2 = 10 / 2 = 5
14. OA = \(\sqrt{OM^2 + AM^2}\) = \(\sqrt{4^2 + 5^2}\) = \(\sqrt{41}\)
15. AB = 2 × OA = 2 × \(\sqrt{41}\)
16. S_ABO = 1/2 × AB × OM = 1/2 × 2√41 × 4 = 4√41 ≈ 25.61
17. S_ABCD = AB × AD = 2√41 × 10 = 20√41 ≈ 128.06
18. S_AMC = 1/2 × AM × CD = 1/2 × 5 × 2√41 = 5√41 ≈ 32.02
19. AD = 8, S_AMC = 24, AM = 2S_AMC / CD = 2 × 24 / AB = 2 × 24 / AB
20. OA = OD, => O - центр прямоугольника, => AO = BO = CO = DO
21. MC = AM, AM = AD / 2 = 8 / 2 = 4
22. AB = CD, => CD = 8, AM = 2S_AMC / CD = 2 × 24 / 8 = 6
23. AB = \(\sqrt{BC^2 + AC^2}\)
24. OM = 1/2 AB, => AB = 2OM = 2 × 3 = 6
25. AD = S_ABCD / AB = 60 / 6 = 10
26. AM = AD / 2 = 10 / 2 = 5
27. S_ABO = 1/2 × AB × OM = 1/2 × 6 × 3 = 9
28. S_AMC = 1/2 × AM × CD = 1/2 × 5 × 6 = 15
29. AB = 7, S_ABO = 21, OM = 2S_ABO / AB = 2 × 21 / 7 = 6
30. AD = 2 × OM = 2 × 6 = 12
31. S_ABCD = AB × AD = 7 × 12 = 84
32. AM = AD / 2 = 12 / 2 = 6
33. S_AMC = 1/2 × AM × CD = 1/2 × 6 × 7 = 21
34. OM = 2, S_AMC = 16, CD = 2S_AMC / AM = 2 × 16 / AM
35. AM = MC, AD = 2 × AM, CD = AB
36. AD = BC, => AM = AD / 2
37. S_ABO = S_AB × OM / 2
38. S_ABCD = AD × AB
39. OM = AM² + AO², AO = 1/2 × AC, AC = \(\sqrt{AD^2 + DC^2}\)