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,
[tex]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}.[/tex]
2. South-East point from the Start: By the 2nd theorem,
[tex]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}.[/tex]
3. West point from the previous: By the 2nd theorem,
[tex]x^2=(32-20)\cdot 32=4\cdot 3\cdot 16\cdot 2\Rightarrow x=2\cdot 4\cdot \sqrt{6}=8\sqrt{6}.[/tex]
4. West point from the previous: By the 1st theorem,
[tex]9^2=x\cdot 15\Rightarrow x=\dfrac{81}{15}=\dfrac{27}{5}=5.4.[/tex]
5. West point from the previous: By the 2nd theorem,
[tex]10^2=8\cdot (8+x)\Rightarrow 8+x=12.5,\ x=4.5.[/tex]
6. North point from the previous: By the 1st theorem,
[tex]x^2=48\cdot 6=6\cdot 4\cdot 2\cdot 6\Rightarrow x=6\cdot 2\cdot \sqrt{2}=12\sqrt{2}.[/tex]
7. East point from the previous: By the 2nd theorem,
[tex]x^2=22.5\cdot 30=225\cdot 3\Rightarrow x=15\sqrt{3}.[/tex]
8. North point from the previous: By the 1st theorem,
[tex]x^2=7.5\cdot 36=270\Rightarrow x=3\sqrt{30}.[/tex]
8. West point from the previous: By the 2nd theorem,
[tex]x^2=12.5\cdot (12.5+13.5)=12.5\cdot 26=25\cdot 13\Rightarrow x=5\sqrt{13}.[/tex]
9. North point from the previous: By the 1st theorem,
[tex]12^2=x\cdot 30\Rightarrow x=\dfrac{144}{30}=4.8.[/tex]
101. East point from the previous: By the 1st theorem,
[tex]6^2=1.6\cdot (x-1.6)\Rightarrow x-1.6=22.5,\ x=24.1.[/tex]
11. East point from the previous: By the 2nd theorem,
[tex]20^2=32\cdot (32-x)\Rightarrow 32-x=12.5,\ x=19.5.[/tex]
12. South-east point from the previous: By the 2nd theorem,
[tex]18^2=x\cdot 21.6\Rightarrow x=15.[/tex]
13. North point=The end.
