given

1. (1)
(3)
(2) (3)

2. (𝜎 𝑛−
𝜎 𝑥 + 𝜎 𝑦
2
)2
+ 𝜏 𝑡
2
= (
𝜎 𝑥 − 𝜎 𝑦
2
)2
+ 𝜏2
(𝜎 𝑛−𝑎)2
+ 𝜏 𝑡
2
= 𝑅2
𝑤ℎ𝑒𝑟𝑒; 𝑎 =
𝜎 𝑥 + 𝜎 𝑦
2
𝑅 = (
𝜎 𝑥 − 𝜎 𝑦
2
)2 + 𝜏2
𝑎 𝑎𝑛𝑑 𝑟𝑎𝑑𝑖𝑢𝑠 𝑅 𝑖𝑛

3. (𝜎 𝑥, 𝜏) 𝜎 𝑥
(𝜎 𝑦, 𝜏) 𝜎 𝑦
=
𝜎 𝑥 + 𝜎 𝑦
2

4. =
𝜎𝑥 − 𝜎 𝑦
2
= (
𝜎 𝑥 − 𝜎 𝑦
2
)2 + 𝜏2
𝜎𝑥 − 𝜎 𝑦
2
= (
𝜎𝑥 − 𝜎 𝑦
2
)2 + 𝜏2

5. (𝜎𝑥, 𝜏)
𝐸(𝜎 𝑦, 𝜏)

6. 𝜎 𝑥
𝜎𝑥 − 𝜎 𝑦
2
𝜏
𝜎𝑥 − 𝜎 𝑦
2
𝜎𝑥 + 𝜎 𝑦
2
𝜏
𝜎 𝑛

7. 𝜎𝑥 − 𝜎 𝑦
2
𝜏
𝜏 𝑡
𝜎 𝑥
𝜎𝑥 + 𝜎 𝑦
2
𝜏 𝑡𝜎 𝑛
𝜎𝑥

8. 𝜎 𝑛
𝜏 𝑡
𝑟

9. 1

10. 𝜎𝑥 𝜎 𝑦 𝜏
𝜎1 𝜎2
𝜏 𝑚𝑎𝑥

12. 2

13. 𝜎𝑥 𝜎 𝑦 𝜏
𝜎1 𝜎2
𝜏 𝑚𝑎𝑥

14. 3

15. 𝜎𝑥 𝜎 𝑦 𝜏
𝜎1 𝜎2
𝜏 𝑚𝑎𝑥
2𝜃1
2𝜃2
2𝜃1 = 72 + 180°
= 252°
2𝜃2 = 72°
or 432°

16. 4

17. 𝜎𝑥 𝜎 𝑦 𝜏
𝜎𝜎 𝑛
𝜏 𝑚𝑎𝑥