The discussion revolves around determining the lengths of the sides of a right-angled triangle when the area and the ratio of the two non-hypotenuse sides are known. Participants explore the relationship between the area and the sides of the triangle, as well as the implications of the given ratio.
The discussion is active, with participants offering insights into the mathematical relationships involved. Some have provided algebraic manipulations that could lead to finding the lengths, while others express uncertainty about the process. There is no explicit consensus yet, but productive lines of reasoning are being explored.
Participants note the challenge of working with the ratio of the sides and the area without knowing the actual lengths. There is also mention of the time constraints affecting the discussion.
tim9000 said:Homework Statement
I know the area of a right angled triangle, I also know the ratio of the two non-hypotenuse sides.
Is there anyway of finding the lengths?
pasmith said:If the sides are [itex]a[/itex] and [itex]b[/itex] then you know that the area of the triangle is [itex]\frac12 ab[/itex].
HallsofIvy said:If you call the two legs a and b then the area is (1/2)ab= A so ab= 2A.
The ratio of the two sides is a/b= r so that a= br. Replace a in the first equation by that: ab= (br)b= rb^2= A so b^2= A/r. Solve for b, then solve for a.