Answer:
See explanation
Step-by-step explanation:
Theorem 1: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.
Theorem 2: The length of each leg of a right triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.
1. Start point: By the 1st theorem,
![x^2=25\cdot (49-25)=25\cdot 24=5^2\cdot 2^2\cdot 6\Rightarrow x=5\cdot 2\cdot \sqrt{6}=10\sqrt{6}.](https://tex.z-dn.net/?f=x%5E2%3D25%5Ccdot%20%2849-25%29%3D25%5Ccdot%2024%3D5%5E2%5Ccdot%202%5E2%5Ccdot%206%5CRightarrow%20x%3D5%5Ccdot%202%5Ccdot%20%5Csqrt%7B6%7D%3D10%5Csqrt%7B6%7D.)
2. South-East point from the Start: By the 2nd theorem,
![x^2=40\cdot (40+5)=4\cdot 5\cdot 2\cdot 9\cdot 5\Rightarrow x=2\cdot 5\cdot 3\cdot \sqrt{2}=30\sqrt{2}.](https://tex.z-dn.net/?f=x%5E2%3D40%5Ccdot%20%2840%2B5%29%3D4%5Ccdot%205%5Ccdot%202%5Ccdot%209%5Ccdot%205%5CRightarrow%20x%3D2%5Ccdot%205%5Ccdot%203%5Ccdot%20%5Csqrt%7B2%7D%3D30%5Csqrt%7B2%7D.)
3. West point from the previous: By the 2nd theorem,
![x^2=(32-20)\cdot 32=4\cdot 3\cdot 16\cdot 2\Rightarrow x=2\cdot 4\cdot \sqrt{6}=8\sqrt{6}.](https://tex.z-dn.net/?f=x%5E2%3D%2832-20%29%5Ccdot%2032%3D4%5Ccdot%203%5Ccdot%2016%5Ccdot%202%5CRightarrow%20x%3D2%5Ccdot%204%5Ccdot%20%5Csqrt%7B6%7D%3D8%5Csqrt%7B6%7D.)
4. West point from the previous: By the 1st theorem,
![9^2=x\cdot 15\Rightarrow x=\dfrac{81}{15}=\dfrac{27}{5}=5.4.](https://tex.z-dn.net/?f=9%5E2%3Dx%5Ccdot%2015%5CRightarrow%20x%3D%5Cdfrac%7B81%7D%7B15%7D%3D%5Cdfrac%7B27%7D%7B5%7D%3D5.4.)
5. West point from the previous: By the 2nd theorem,
![10^2=8\cdot (8+x)\Rightarrow 8+x=12.5,\ x=4.5.](https://tex.z-dn.net/?f=10%5E2%3D8%5Ccdot%20%288%2Bx%29%5CRightarrow%208%2Bx%3D12.5%2C%5C%20x%3D4.5.)
6. North point from the previous: By the 1st theorem,
![x^2=48\cdot 6=6\cdot 4\cdot 2\cdot 6\Rightarrow x=6\cdot 2\cdot \sqrt{2}=12\sqrt{2}.](https://tex.z-dn.net/?f=x%5E2%3D48%5Ccdot%206%3D6%5Ccdot%204%5Ccdot%202%5Ccdot%206%5CRightarrow%20x%3D6%5Ccdot%202%5Ccdot%20%5Csqrt%7B2%7D%3D12%5Csqrt%7B2%7D.)
7. East point from the previous: By the 2nd theorem,
![x^2=22.5\cdot 30=225\cdot 3\Rightarrow x=15\sqrt{3}.](https://tex.z-dn.net/?f=x%5E2%3D22.5%5Ccdot%2030%3D225%5Ccdot%203%5CRightarrow%20x%3D15%5Csqrt%7B3%7D.)
8. North point from the previous: By the 1st theorem,
![x^2=7.5\cdot 36=270\Rightarrow x=3\sqrt{30}.](https://tex.z-dn.net/?f=x%5E2%3D7.5%5Ccdot%2036%3D270%5CRightarrow%20x%3D3%5Csqrt%7B30%7D.)
8. West point from the previous: By the 2nd theorem,
![x^2=12.5\cdot (12.5+13.5)=12.5\cdot 26=25\cdot 13\Rightarrow x=5\sqrt{13}.](https://tex.z-dn.net/?f=x%5E2%3D12.5%5Ccdot%20%2812.5%2B13.5%29%3D12.5%5Ccdot%2026%3D25%5Ccdot%2013%5CRightarrow%20x%3D5%5Csqrt%7B13%7D.)
9. North point from the previous: By the 1st theorem,
![12^2=x\cdot 30\Rightarrow x=\dfrac{144}{30}=4.8.](https://tex.z-dn.net/?f=12%5E2%3Dx%5Ccdot%2030%5CRightarrow%20x%3D%5Cdfrac%7B144%7D%7B30%7D%3D4.8.)
101. East point from the previous: By the 1st theorem,
![6^2=1.6\cdot (x-1.6)\Rightarrow x-1.6=22.5,\ x=24.1.](https://tex.z-dn.net/?f=6%5E2%3D1.6%5Ccdot%20%28x-1.6%29%5CRightarrow%20x-1.6%3D22.5%2C%5C%20x%3D24.1.)
11. East point from the previous: By the 2nd theorem,
![20^2=32\cdot (32-x)\Rightarrow 32-x=12.5,\ x=19.5.](https://tex.z-dn.net/?f=20%5E2%3D32%5Ccdot%20%2832-x%29%5CRightarrow%2032-x%3D12.5%2C%5C%20x%3D19.5.)
12. South-east point from the previous: By the 2nd theorem,
![18^2=x\cdot 21.6\Rightarrow x=15.](https://tex.z-dn.net/?f=18%5E2%3Dx%5Ccdot%2021.6%5CRightarrow%20x%3D15.)
13. North point=The end.