One leg of a right triangle is one-third as long as the other leg, and the perimeter is 10. Find the lengths of all three sides, to three decimal places.

Question

calculusxy · Answer

@jim_thompson5910

jim_thompson5910 · Answer

show me what you have so far

yournightmare · Answer

well im pretty sure one of them would be 2. the others would probably be 5 and 3.

calculusxy · Answer

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sshayer · Answer

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jim_thompson5910 · Answer

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calculusxy · Answer

To find out c I did 
$\sqrt{(\frac{1}{3}x)^2 + x^2}$

jim_thompson5910 · Answer

good

jim_thompson5910 · Answer

so the three sides are some expression in terms of x. Add up those three sides to get the given perimeter. Then solve for x

calculusxy · Answer

So I thought of distributing the x's to find the squares of the terms under the square root and got 
$$\sqrt{\frac{1}{9}x^2 + x^2} = \sqrt{\frac{10}{9}x^2} = c$$

jim_thompson5910 · Answer

That works

calculusxy · Answer

And then I have now $$\frac{1}{3}x + x + \sqrt{\frac{10}{9}x^2} = 10$$
$$\frac{4}{3}x + \sqrt{\frac{10}{9}x^2} = 10$$

Is it correct so far?

jim_thompson5910 · Answer

so far, so good

will.h · Answer

And solve x. It is

calculusxy · Answer

$$(\frac{4}{3}x + \sqrt{\frac{10}{9}x^2})^2 = 10^2$$
$$\frac{16}{9}x^2 + \frac{10}{9}x^2 = 10$$

Correct so far?

jim_thompson5910 · Answer

incorrect

jim_thompson5910 · Answer

you need to use the box method or FOIL for the left side

on the right side, 10^2 turns into 100

calculusxy · Answer

The right side was a typo sorry

calculusxy · Answer

But I'll fix the left side

calculusxy · Answer

Will the left side be something like
$$\frac{16}{9}x^2 + \frac{2}{3}x \sqrt{\frac{10}{9}x^2} + \frac{10}{9}x^2 = 100$$

jim_thompson5910 · Answer

I don't agree with the middle term on the left side

calculusxy · Answer

I am sorry I meant to write 8/3

jim_thompson5910 · Answer

good

jim_thompson5910 · Answer

change the 2/3 to 8/3 and leave everything else in that middle term the same

jim_thompson5910 · Answer

also you can take advantage of the fact that

sqrt(x^2) = x

assuming x >= 0

calculusxy · Answer

$$\frac{26}{9}x^2 + \frac{8}{3}x^2 \sqrt{\frac{10}{9}x} = 100$$

jim_thompson5910 · Answer

x won't stay in the square root after you use the rule `sqrt(x^2) = x`

calculusxy · Answer

Oh yes totally forgot about that

calculusxy · Answer

$$\frac{26}{9}x^2 + \frac{8}{3}x^2 \sqrt{\frac{10}{9}} = 100$$

jim_thompson5910 · Answer

good

jim_thompson5910 · Answer

then you can factor out the GCF x^2

calculusxy · Answer

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