# Find the length of base of a triangle without using Pythagorean Theorem I'm curious whether it is p

Find the length of base of a triangle without using Pythagorean Theorem
I'm curious whether it is possible to find the length of base of the triangle without using Pythagorean Theorem
No Pythagorean Theorem mean:
=> No trigonometric because trigonometric is built on top of Pythagorean Theorem. etc $\mathrm{sin}\theta =\frac{a}{r}$
=> No Integration on line or curve because the integration is built on top of Pythagorean Theorem. etc: $s\left(x\right)=\int \sqrt{{f}^{\prime }\left(x{\right)}^{2}+1}$
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stafninumfu1tf
It is perfectly possible to construct this figure in a way such that ${a}_{1}\ne a.$.
But in the case where ${a}_{1}=a,$, we can observe that the perpendicular line labeled g divides the large triangle into two congruent triangles, hence the triangle with hypotenuse AB is twice the area of the triangle with hypotenuse a.
The larger triangle also is similar to the smaller triangle. In order to have twice the area, all its dimensions, including the length of the hypotenuse, must therefore be greater by a factor of $\sqrt{2}.$.
A somewhat more complicated version of this argument can be applied in the case where ${a}_{1}\ne a,$, resulting in a proof of the Pythagorean Theorem. I would argue that proving the Pythagorean Theorem is technically not the same thing as using it but would still be a violation of the spirit of the question. Proving only the special case ${a}_{1}=a,$, however, does not rise to the level of a proof of the Pythagorean Theorem (which is more general).