Right triangles and ratios

[dwill]

In right triangle ABC, the ratio of the legs is 2 to 3. If the area of traingle ABC is 75, what is the length of hypotenuse AC?


I tried to solve this using the pythogorian equation, with 2 and 3 as a and b. it did not work.

I'm not sure how I am supposed to use area to solve this question.

 

[soroban]

Hello, dwill![/soze]

In right triangle ABC, the ratio of the legs is 2 to 3.
If the area of traingle ABC is 75, what is the length of hypotenuse AC?

The legs are in the ratio 2:3.
This means that: \(\displaystyle a\,=\,2n,\;b\,=\,3n\,\) for some integer \(\displaystyle n\).

The area is: \(\displaystyle \:A \:=\:\frac{1}{2}bh\;\;\Rightarrow\;\;\frac{1}{2}(2n)(3n} \:=\:75\;\;\Rightarrow\;\;n^2\,=\,25\;\;\Rightarrow\;\;n\:=\:5\)

Hence, the two legs are: \(\displaystyle \:a\,=\,10,\:b\,=\,15\)

Therefore, the hypotenuse is: \(\displaystyle \:\sqrt{10^2\,+\,15^2}\:=\:\sqrt{325} \:=\:5\sqrt{13}\)