# Arithmetic sequence and a rectangular triangle

Task: Calculate the smallest angle in a rectangular (or right) triangle whose sides form an arithmetic sequence.

That's it.. and I can't solve it. The solution is 36°52'.

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We can set the sides to be $a-d,a,a+d$

So we have $(a-d)^2+a^2=(a+d)^2\iff a^2=4ad\iff a=4d$ as $a>0$

So, the sides become $3d,4d,5d$

The smallest angle is due to the smallest side

So, if $A$ is the smallest angle $\displaystyle\sin A=\frac{3d}{5d}=\frac35$

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wow thank you soo much!! You're fast! :) – cherry8.8vanilla Sep 30 '13 at 14:57
@Anja97, my pleasure. Hope I could make the idea clear – lab bhattacharjee Sep 30 '13 at 14:59
Yes, you sure did :) It's exactly what I was looking for :) thank you again! – cherry8.8vanilla Sep 30 '13 at 15:34
I find it interesting that this shows that the 3-4-5 right triangle is the only one (aside from scaling) with sides in an arithmetic sequence. – marty cohen Sep 30 '13 at 16:34

It seems that you're looking at a right triangle with integer side lengths. The side lengths for these triangles are called Pythagorean triples. Look at the site I linked to. Which Pythagorean triple has small values and is an arithmetic sequence? (The angle is more accurately approximately $36^\circ52'11.63''$.)

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