MHB Can Pythagoras' Theorem Prove this Right Triangle Hypotenuse Equation?

AI Thread Summary
The discussion revolves around proving the hypotenuse equation for a right triangle using Pythagoras' Theorem. Given the legs of the triangle as u and v, with specific expressions for each, the goal is to show that w equals a derived formula involving m and n. The participants agree that squaring u and v is necessary to demonstrate the relationship w = [2(m^2 + n^2)/(m - n)n. The calculations involve substituting the expressions for u and v into the Pythagorean identity and simplifying to find an identity. The conversation concludes with the affirmation of proceeding with the calculations to achieve the proof.
mathdad
Messages
1,280
Reaction score
0
A right triangle is given. One leg is u units and the other leg is v units. The hypotenuse is given to be w units.

If u = [2(m + n)]/n and v = 4m/(m - n), show that
w = [2(m^2 + n^2)/(m - n)n.

Must I square u and v to show that
w = [2(m^2 + n^2)/(m - n)n?
 
Mathematics news on Phys.org
According to Pythagoras, we should have:

$$u^2+v^2=w^2$$

Plug in the given values:

$$\left(\frac{2(m+n)}{n}\right)^2+\left(\frac{4m}{m-n}\right)^2=\left(\frac{2(m^2+n^2)}{(m-n)n}\right)^2$$

I would multiply through by $$(n(m-n))^2$$ to get:

$$(2(m+n)(m-n))^2+(4mn)^2=(2(m^2+n^2))^2$$

$$(2(m^2-n^2))^2+(4mn)^2=(2(m^2+n^2))^2$$

Continuing, you will obtain an identity. :D
 
MarkFL said:
According to Pythagoras, we should have:

$$u^2+v^2=w^2$$

Plug in the given values:

$$\left(\frac{2(m+n)}{n}\right)^2+\left(\frac{4m}{m-n}\right)^2=\left(\frac{2(m^2+n^2)}{(m-n)n}\right)^2$$

I would multiply through by $$(n(m-n))^2$$ to get:

$$(2(m+n)(m-n))^2+(4mn)^2=(2(m^2+n^2))^2$$

$$(2(m^2-n^2))^2+(4mn)^2=(2(m^2+n^2))^2$$

Continuing, you will obtain an identity. :D

Will do so. Thanks.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top