Mr.Math
  • Mr.Math
\(\large \large \text{ Fun With Mr.Math #2}\) Determine the side lengths of a right triangle if they are integers and the product of the legs' lengths equals three times the perimeter.
Meta-math
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
jamiebookeater
  • jamiebookeater
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
asnaseer
  • asnaseer
|dw:1328658036624:dw| \[\begin{align} ab&=3(a+b+c)\tag{a}\\ &\qquad\text{rearranging (a) we get:}\\ 3c&=ab-3(a+b)\\ \therefore 9c^2&=a^2b^2-6ab(a+b)+9(a^2+b^2+2ab)\tag{b}\\ &\qquad\text{from Pythagorus we get:}\\ a^2+b^2&=c^2\\ \therefore 9a^2+9b^2&=9c^2\\ &=a^2b^2-6ab(a+b)+9(a^2+b^2+2ab)\hspace{2cm}\text{[using (b)]}\\ \therefore ab&=6(a+b-3)\tag{c}\\ &\qquad\text{substituting (c) into (a) we get:}\\ 6(a+b-3)&=3(a+b+c)\tag{d}\\ \therefore a+b-c&=6 \end{align}\] The solution is therefore al Pythagorean triples satisfying (d), which leads to these unique solutions: a=7, b=24, c=25 a=8, b=15, c=17 a=9, b=12, c=15 We can get 3 more solutions by swapping the values for a and b but I believe that these would not add anything new.
Mr.Math
  • Mr.Math
Very good and neat asnaseer, as always! :-)

Looking for something else?

Not the answer you are looking for? Search for more explanations.