Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The question:

Suppose there is a right triangle with sides $a$ and $b$ and hypotenuse $c$. Its perimeter is the same as its area, and $b = 6$. What are its side lengths?

I just cannot figure out how to do this! The second sentence isn't particularly helpful:

$$a + c + 6 = \frac{6a}{2}$$ $$a + c + 6 = 3a$$ $$???$$

And I can't get anywhere with the Pythagorean Theorem either:

$$a^2 + 36 = c^2$$ $$c^2 - 36 = a^2$$ $$(c + 6)(c - 6) = a^2$$ $$???$$

How do I solve this puzzle?

share|cite|improve this question
Do you have an error with your formula? if a =6, why should you mention a and 6? Do you imply that b is also = 6. IF a = 6, then you can use this fact to have only 2 unknowns. That should simplify the formulas somewhat. – cuabanana Sep 11 '13 at 23:14
You seem to have set $b = 6$ on the LHS of the first equation. Do so also on the right. – Daniel Fischer Sep 11 '13 at 23:14
@cuabanana Whoops, typo. I meant $b = 6$. – Doorknob Sep 11 '13 at 23:15
@DanielFischer Okay, that still didn't help :/ – Doorknob Sep 11 '13 at 23:16
You get $a = \frac{c+6}{2}$. So $c^2 = 36 + \frac{(c+6)^2}{4}$. $4(c+6)(c-6) = (c+6)^2$, $4(c-6) = (c+6), 3c = 30$. – Daniel Fischer Sep 11 '13 at 23:22
up vote 2 down vote accepted

If $a=6$, then area is equal to $(6b)/2=3b$. If that is equal to the perimeter, then $a+b+c=3b$ so $a+c=2b$ so $6+c=2b$. You also know that $a^2+b^2=c^2$ so $36+b^2=c^2$.

From the $6+c=2b$ you get $c=2b-6$, and so $36+b^2=(2b-6)^2$. Expand and you get a quadratic on $b$.

share|cite|improve this answer
Sorry, I made a typo. I meant $b = 6$, not $a = 6$. – Doorknob Sep 11 '13 at 23:17
Its the same. Just change $a$ for $b$ and $b$ for $a$ everywhere above lol – Daniel Montealegre Sep 11 '13 at 23:23
Oh, alright, well thanks then! +1 – Doorknob Sep 11 '13 at 23:26

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.