SOLUTION: Given 30-60-90 triangle with sides p\sqrt(6) , p\sqrt(2 ), q\sqrt(3); find p, q

Question 1205030: Given 30-60-90 triangle with sides p\sqrt(6) , p\sqrt(2 ), q\sqrt(3); find p, q

Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13206)   (Show Source): You can put this solution on YOUR website!

Don't use "\" -- it might have a special meaning in some area(s) of mathematics.

I assume the side lengths are p/sqrt(6), q/sqrt(3), and p/sqrt(2).

The squares of the side lengths are then p^2/6, q^2/3, and p^2/2.

On first glance, with the denominators 6, 3, and 2, I immediately see that, if p and q are both equal to the same number x, then I have x^2/6+x^2/3=x^2/2, which is true for all values of x.

So the problem has an infinite number of solutions in which p=q.

But there might be other solutions hiding somewhere, so lets' look at the problem more closely.

We know that p/sqrt(2) is greater than p/sqrt(6); but q/sqrt(3) could be less than or greater than p/sqrt(2). So there are two cases to consider: the longest side (hypotenuse) can be either p/sqrt(2) or q/sqrt(3).

Case 1: the hypotenuse is p/sqrt(2)

(Note this is the case discussed informally above.)

Case 2: The hypotenuse is q/sqrt(3)

This case also has an infinite number of solutions, where p is any number x and q is x*sqrt(2).

ANSWER:
p = any number;
q = p OR q=p*sqrt(2)

Answer by ikleyn(52865)   (Show Source): You can put this solution on YOUR website!
.
Given 30-60-90 triangle with sides p/sqrt(6) , p/sqrt(2 ), q/sqrt(3); find p, q
~~~~~~~~~~~~~~~~~~~~~

        Notice that I rewrote the condition using normal designations for the division operation.

Two sides and have the ratio = = . In combination with the given fact that the triangle is 30-60-90, it means that is the shorter leg, while is the longer leg. +-----------------------------------------------+ | It admits only one interpretation, | | and no other interpretation is admittable. | +-----------------------------------------------+ Hence, is the hypotenuse, and we can write = (the hypotenuse length is twice the shorter leg), since the triangle is 30-60-90. It gives q = = = . So, the sides are: is the shorter leg, is the longer leg and = is the hypotenuse. We can write then Pythagorean equation + = , or + = , + = , = , = , which is an identity. At this point, the problem is solved in full. ANSWER. p can be any real positive number; then q = . There are infinitely many solutions.

Solved.

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