Draft:Dylan Right Triangle Theorem

• Comment: Fails WP:GNG, lacks any sources. Dan arndt (talk) 03:03, 1 February 2024 (UTC)

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The Dylan Right Triangle Theorem was created by Dylan Sood, a geometry student, with help from Raphael Chu, another geometry student. The Dylan Right Triangle Theorem shows that for any leg in a right triangle, you can find another leg and the hypotenuse that would work for a right triangle. This theorem is important for math competitions, geometry, real world applications, and is helpful for your math journey. Using this will allow you to find a leg and hypotenuse when given only one leg. This works for any right triangle, so long that you use a leg and not the hypotenuse. You can use this for pythagorean triplets, when you know a single number.

Theorem

When you are given one right triangle, and the leg X you can use the Dylan Right Triangle Theorem. You can proof this from the pythagorean theorem, but you can use this with only one number and without using the pythagorean theorem.

Given
△ABC is a right triangle
m∠B = 90°
${\overline {AB}}$ = $x$

Dylan Right Triangle Theorem States:
${\overline {AB}}$ = $x$
${\overline {BC}}$ = ${\frac {x^{2}-1}{2}}$
${\overline {AC}}$ = ${\frac {x^{2}+1}{2}}$

Proof
1. $(x)^{2}+({\frac {x^{2}-1}{2}})^{2}=({\frac {x^{2}+1}{2}})^{2}$

2. $x^{2}+{\frac {(x^{2}-1)^{2}}{4}}={\frac {(x^{2}+1)^{2}}{4}}$

3. ${\frac {4x^{2}}{4}}+{\frac {x^{4}-2x^{2}+1}{4}}={\frac {x^{4}+2x^{2}+1}{4}}$

4. ${\frac {x^{4}-2x^{2}+4x^{2}+1}{4}}={\frac {x^{4}+2x^{2}+1}{4}}$

5. ${\frac {x^{4}+2x^{2}+1}{4}}={\frac {x^{4}+2x^{2}+1}{4}}$

This shows that both sides are equal to eachother. Hence, proving the theorem.